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\centerline {{\bf Limit theorems and infinitely divisible
distributions.} \ \ {\rm  Part III}}
\centerline{\it Functional limit theorems.}
\smallskip
\centerline{\it by P\'eter Major}
\smallskip
\centerline{The Mathematical Institute of the Hungarian Academy of
Sciences} \medskip
 
\noindent
{\narrower {\narrower
{\it Summary:}\/ In the third part of this work we prove the
sufficiency part of the main result proved in the second part with a
different method. We make a good coupling of the independent random
variables for whose sums we want to prove a limit theorem with
independent random variables with so-called associated distributions to
the distributions of the original random variables. This coupling shows
that the sums of the original random variables and the sums of the
random variables with these associated distributions have the same
limit behaviour. This proof helps us to understand better the picture
behind the limit theorem we discuss. Beside this, its method enables us
to prove a functional limit theorem version of this result. \par}\par}
 
\beginsection 1. Introduction. Formulation of the results.
 
In Theorem~1 of Part~II of the work {\it Limit theorems and infinitely
divisible distributions}\/ we gave the necessary and sufficient
condition for the convergence in distribution of the appropriate
normalized sums of the random variables in a triangular array which
satisfies the uniform smallness condition. We recall the sufficiency
part of this result and give a new proof of this result which is
based on a good coupling. Then, by applying this coupling argument
again, we also prove a functional limit theorem version of this result.
This functional limit theorem, whose exact formulation will be given
later, states that under natural weak conditions not only the sums of
the random variables in a row have a limit distribution, but also the
distributions of the random broken lines, made from the partial sums in
a natural way converge in distribution to a probability measure in the
space of functions.
 
We will investigate the following result of Part~II.
\medskip\noindent
{\bf Theorem~1.} {\it Let $\xi_{k,j}$, be a triangular array with
distribution functions $F_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
which satisfies the uniform smallness condition. Let us introduce
the canonical measures
$$
M_k(\,dx)=\sum_{j=1}^{n_k}x^2 F_{k,j}(\,dx), \quad k=1,2,\dots, \tag1.1
$$
fix some number $a>0$, and define the function
$$
\tau(x)=\tau_a(x)=\cases
x&\text{if }|x|\le a\\
a&\text{if }x\ge a  \\
-a&\text{if }x\le -a
\endcases\;.  \tag1.2
$$
Let us assume that the random variables $\xi_{k,j}$ satisfy the identity
$E\tau(\xi_{k,j})=0$ for all indices $k=1,2,\dots$, $1\le j\le n_k$,
and that the measures $M_k$ converge to a canonical measure $M_0$ on
the real line.
 
Then the sums $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ converge in distribution
to a distribution function whose characteristic function $\varphi(t)$, or
more explicitly its (existing) logarithm, is given by the formula
$$
\log \varphi(t)=\int_{-\infty}^\infty \frac{e^{itu}-1-it\tau(u)}{u^2}
M_0(\,du), \tag1.3
$$
where the canonical measure $M_0$ is the limit of the canonical measures
$M_k$, and the function $\tau$ was defined in formula (1.2).} \medskip

To understand better the above result we recall some notions. A
$\sigma$-finite measure $M$ on the real line is called canonical if
$$
M([-a,a])<\infty,\quad \text{and} \quad \int_{\{|x|>a\}}
\frac1{x^2}M(\,dx)<\infty
$$
for all real numbers $a>0$. A sequence of canonical measures $M_k$,
$k=1,2,\dots$, (weakly) converges to a canonical measure $M_0$ if
$$
\aligned
\lim_{k\to\infty}M_k^+(x)&=
\lim_{k\to\infty}\int_x^\infty\frac1{u^2}M_k(\,du)=
M_0^+(x)=\int_x^\infty\frac1{u^2}M_0(\,du), \\
\lim_{k\to\infty}M_k^-(x)&=
\lim_{k\to\infty}\int_{-\infty}^{-x}\frac1{u^2}M_k(\,du)
=M_0^-(x)=\int_{-\infty}^{-x}\frac1{u^2}M_0(\,du),
\endaligned \tag1.4
$$
for all such numbers $x>0$ where the function $M_0^+(\cdot)$ or
$M_0^-(\cdot)$ is continuous, and
$$
\lim_{k\to\infty} M_k\{[a,b]\}=M_0\{[a,b]\}
$$
for all numbers $-<\infty<a<b<\infty$ where the limit measure $M_0$
is continuous. (The continuity of the measure $M_0$ in the points $a$
and $b$  means that $M_0(\{a\})=M_0(\{b\})=0$).
 
A triangular array $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, consists
of random variables from which the random variables with fixed
first index~$k$ are independent. It satisfies the uniform smallness
condition if for all $\e>0$ the relation $\limm_{k\to\infty}\supp_{1\le
j\le n_k}P(|\xi_{k,j}|>\e)=0$ holds.
 
To give a complete formulation of Theorem~1, or more explicitly of
formula~(1.3) we have to define the integrand in the integral of
formula~(1.3) also in the point $u=0$. We  do this by continuity
arguments in the following way:
$$
\left.\frac{e^{itu}-1-it\tau(u)}{u^2}\right|_{u=0}=
\lim_{u\to0}\frac{e^{itu}-1-it\tau(u)}{u^2}=-\frac{t^2}2.
$$
 
\headline{\ifodd\pageno \hfill {\it Limit theorems and infinitely
divisible distributions.} {\rm Part III} \hfill \else
\hfill {\it P\'eter Major} \hfill \fi}
 
The above formulated Theorem~1 contains only one part of Theorem~1 in
Part~II of this  work, the sufficiency part which stated the existence
of a limit distribution if the conditions of the theorem are satisfied.
Moreover, even this  result is formulated in the special case when the
condition $E\tau(\xi_{k,j})=0$ holds for all indices $k=1,2,\dots$ and
$1\le j\le n_k$. But the general case can be reduced  simply to  this
special case with the help of Lemma~2 in Part~II. The formulation of
the result is simpler in the special case considered here. Now the more
complicated conditions about the behaviour of the measures~$M_k$ can be
expressed as the weak convergence of the canonical measures $M_k$
introduced in Theorem~1 to an appropriate canonical measure $M_0$.
 
In the problems of probability theory we often take from the terms
of the random sum we investigate the expected value of these terms and
in such a way we work with the sum of random variables with expectation
zero. The condition $E\tau(\xi_{k,j})=0$ is a modified version of this
property in the general case when the random variables we are working
with may not have a finite expectation.
 
The proof of Theorem~1 applies a method essentially different from the
method of Part~II. We shall apply the following relatively simple result
whose proof will be given at the end in the Appendix.
\medskip\noindent
{\bf Theorem~A.} {\it Let $S_k$ and $T_k$, $k=1,2,\dots$, be sequences
of such random variables for which the sequence of differences
$S_k-T_k$ stochastically converges to zero as $k\to\infty$. If the
sequence of random variables $S_k$ converges in distribution to a
distribution function~$F$, then the sequence of random variables $T_k$
converges in distribution to the same distribution function~$F$.
 
Also the following generalization of this statement holds. Let a
separable metric space $(X,\rho)$ be given together with two
sequences of random variables $S_k$ and $T_k$, $k=1,2,\dots$,
on a probability space which take their values on the space
$(X,\rho)$, and their distance
$\rho(S_k,T_k)$ tends to zero stochastically if $k\to\infty$. It the
sequence $S_k$ converges weakly to a measure $\mu$ on the space
$(X,\rho)$, then the sequence of random variables $T_k$ converges weakly
to the same measure~$\mu$.}
 
In the proof of Theorem~1 we only need the first statement of Theorem~A.
Its second statement is formulated because that will be needed in the
proof of the functional limit theorem version of Theorem~1. The proof
of Theorem~1 given here will be similar to the second proof of the
Poisson limit theorem in the Appendix of Part~I. We shall make a
good coupling of the random variables considered in Theorem~1 with
independent random variables with infinitely divisible distributions.
We can make this coupling in such a way that the sums of the original
and the sums of the coupled random variables are close to each other,
and the sums of the counting measures of the Poisson measures whose
(normalized) sums determine the infinitely divisible random variables
we construct in this coupling construction is also convergent. Then
Theorem~A enables us to reduce the proof of Theorem~1 to the proof of
the convergence of the coupled random variables in distribution. This
latter statement can also be proved by means of a good coupling
procedure and Theorem~A.
 
To carry out the program sketched above it is useful to decompose the
measures $M_k$ appearing in the formulation of Theorem~1 to two terms
in such a way that the first term is responsible for the convergence of
the Gaussian and the second term for the convergence of the Poissonian
part in the limit theorem we consider. The subsequent Lemma~1 supplies
such a decomposition.

\medskip\noindent
{\bf Lemma~1.} {\it Let $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le
n_k$, be a triangular array satisfying the uniform smallness condition.
Let $F_{k,j}$ denote the distribution function of the random variable
$\xi_{k,j}$, and put $G_{k,j}(\,dx)=x^2F_{k,j}(\,dx)$. Let us assume that
the canonical measures $M_k=\summ_{j=1}^{n_k}G_{k,j}$ converge weakly
to a canonical measure~$M_0$ as $k\to\infty$. Let us write the limit
measure $M_0$ in the form $M_0=M'_0+M''_0$, where  $M'_0$ is the
restriction of the measure~$M_0$ to the origin, i.e.\ for all
measurable sets $A\subset R^1$ $M'_0(A)=0$, if $0\notin A$, and
$M'_0(A)=M_0(\{0\})$ if $0\in A$. Furthermore $M''_0=M_0-M'_0$. Then
there exists such a sequence of numbers $\e_k>0$, $\e_k\to0$ if
$k\to\infty$, for which the measures $M'_k$, $M'_k(A)=M_k(A\cap
I(\{|x|<\e_k\})$ converge weakly to the canonical measure $M'_0$, and
the canonical measures $M''_k$, $M''_k(A)=M_k(A\cap I\{|x|\ge\e_k\})$
converge weakly to the canonical measure $M''_0$, where $I(B)$ denotes
the indicator function of the set~$B$. Furthermore,
$\limm_{k\to\infty}\supp_{1\le j\le n_k}(1-F_{k,j}(\e_k))=0$, and
$\limm_{k\to\infty}\supp_{1\le j\le n_k}F_{k,j}(-\e_k)=0$, and even the
relation
$$
\lim_{k\to\infty}\sum_{j=1}^{n_k}
[(1-F_{k,j}(\e_k))+F_{k,j}(-\e_k)]^2=0 \tag1.5
$$
holds.} \medskip
 
We describe the coupling construction which enables the reduction of
Theorem~1 to the proof of the convergence of appropriate infinitely
divisible distributions. Then we shall also give the ideas behind this
construction and formulate Proposition~1 which tells the most
important properties of this coupling construction. Because of some
reason of convenience we shall work in the coupling construction not
with the random variables $\xi_{k,j}$ considered in Theorem~1, but we
shall construct instead new (for fixed index~$k$ independent)
random variables with the same distributions.
 
Let the random variables $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
of the triangular array we consider satisfy the conditions of
Theorem~1. For all pairs of indices $k=1,2,\dots$, $1\le j\le n_k$,
let us consider the random variables $\xi_{k,j}$  and define the
probability measures $\bar\nu_{k,j}$ and $\bar{\bar\nu}_{k,j}$ given
by the formulas $\bar\nu_{k,j}(A)=P(\xi_{k,j}\in A|\,|\xi_{k,j}|<\e_k)$
and $\bar{\bar\nu}_{k,j}(A)=P(\xi_{k,j}\in A|\,|\xi_{k,j}|\ge \e_k)$
together with the numbers $p_{k,j}=P(|\xi_{k,j}|\ge\e_k)$ where
$A\in\bold R^1$ are arbitrary measurable sets and the numbers $\e_k$
are chosen in such a way that the results of Lemma~1 hold with them.
Let $\eta'_{k,j}$, $j=1,\dots,n_k$, be  $\bar\nu_{k,j}$ distributed
random variables which are independent for fixed index~$k$. Let us
also consider a sequence of Poisson distributed random variables
$\zeta_{k,j}$ with parameters $\bar p_{k,j}$, $k=1,2,\dots$, $1\le j\le
n_k$, where the number $\bar p_{k,j}$ is the solution of the equation
$1-e^{-\bar p_{k,j}}=p_{k,j}$. Let us also assume that these random
variables $\zeta_{k,j}$ are independent for a fixed index $k$, and
they are also independent of the random variables $\eta'_{k,j}$.
Furthermore, let $\gamma_{k,j,l}$, $k=1,2,\dots$, $1\le j\le n_k$,
$l=1,2,\dots$, be random variables with distribution
$\bar{\bar\nu}_{k,j}$ which are independent both from
each other and the random variables defined before.
 
Let us define, beside the already constructed random variables
$\eta'_{k,j}$ the random variables
$\eta''_{k,j}=\summ_{l=1}^{\zeta_{k,j}}\gamma_{k,j,l}$,
$\xi'_{k,j}=\eta'_{k,j}I(\zeta_{k,j}=0)$,
$\xi''_{k,j}=\gamma_{k,j,1}I(\zeta_{k,j}\ge1)$,
$\tilde\xi_{k,j}=\xi'_{k,j}+\xi''_{k,j}$, and
$\eta_{k,j}=\eta'_{k,j}+\eta''_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$.
We shall see that the random variables $\tilde\xi_{k,j}$ and
$\eta_{k,j}$ constructed in such a way give a good coupling which
satisfy Proposition~1 formulated below. This enables us to reduce the
proof of Theorem~1 to the study of the sums of the random variables
$\eta_{k,j}$ which is a simpler problem.
 
The idea behind the above construction is the following. As we shall
see the random variables $\xi_{k,j}$ and
$\tilde\xi_{k,j}=\xi'_{k,j}+\xi''_{k,j}$ have the same distributions.
The random variables $\xi'_{k,j}$ and $\eta'_{k,j}$ are close to each
other, hence they satisfy the central limit theorem with a Gaussian
limit with the same expected value and variance. The reason we defined
them in a slightly different way is that we wanted to achieve that the
random variables $\eta'_{k,j}$ and $\eta''_{k,j}$ be independent,
because this allows to study their behaviour separately. The random
variables $\xi''_{k,j}$ and $\eta''_{k,j}$ are also sufficiently close
to each other, but this closeness has a different reason.We can observe
that both probabilities $P(\eta'_{k,j}\neq0)$ and
$P(\eta''_{k,j}\neq0)$ are small for large indices~$k$, but we need
more knowledge about their behaviour. Our construction guarantees that
$\xi''_{k,j}(\oo)=\eta''_{k,j}(\oo)$ on the set
$\{\oo\colon\;\zeta_{k,j}(\oo)\le1\}$, and the set
$\{\oo\colon\; \zeta_{k,j}(\oo)\ge2\}$ has very small probability
for large indices~$k$. This fact will guarantee that the above
constructed coupling is good for our purposes.
 
Let us also observe that the sequence of random variables
$\gamma_{k,j,1},\dots,\gamma_{k,j,\zeta_{k,j}}$ (with a random number of
elements) is a Poisson process with counting measure
$\bar p_{k,j}\bar{\bar\nu}_{k,j}$. Hence, as we have seen in Part~I 
the random variable $\eta''_{k,j}=\summ_{l=1}^{\zeta_{k,j}}\gamma_{k,j,l}$
is infinitely divisible, and the logarithm of its characteristic
function can be given by the formula
$$
\log \varphi_{k,j}(t)=\log Ee^{it\eta''_{k,j}}=\int
(e^{itu}-1)p_{k,j}\bar{\bar\nu}_{k,j}(\,du)=\frac{\bar p_{j,k}}{ p_{j,k}}
\int_{\{|u|\ge\e_k\}}\frac{e^{itu}-1}{u^2} G_{k,j}(\,du), \tag1.6
$$
where $G_{k,j}(\,du)=u^2F_{k,j}(\,du)$ agrees with the measure $G_{k,j}$
defined in Lemma~1.
 
We formulate the most important properties of the above constructed
coupling in the following Proposition~1.
 
\medskip\noindent
{\bf Proposition~1.} {\it Let the triangular array $\xi_{k,j}$,
$k=1,2,\dots$, $1\le j\le n_k$, satisfy the conditions of Theorem~1.
Then the above coupling construction formulated after Lemma~1 has
the following properties:  The distribution of the random
variables $\tilde\xi_{k,j}=\xi_{k,j}'+\xi_{k,j}''$ and
$\xi_{k,j}$ agree. The triangular arrays $\eta'_{k,j}$ and
$\eta''_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$ are independent, i.e.\
for a fixed index $k$ the random vectors $\eta'_{k,j}$, $1\le j\le
n_k$, and $\eta''_{k,j}$, $1\le j\le n_k$, are independent.
Also the identities $P(|\xi'_{k,j}|\le\e_k)=P(|\eta'_{k,j}|\le\e_k)=1$
hold with the same sequence of numbers $\e_k$, $k=1,2,\dots$, which
appears in Lemma~1. Furthermore,
$$
\limm_{k\to\infty}\summ_{j=1}^{n_k}|E\xi'_{k,j}-E\eta'_{k,j}|=0, \tag1.7
$$
and also the relations
$$
\aligned
&\sup_{1\le p\le n_k}\left|\sum_{j=1}^p(\xi'_{k,j}-E\xi'_{k,j})
-(\eta'_{k,j}-E\eta'_{k,j})\right|\Rightarrow0, \\
&\sup_{1\le p\le n_k}\left|\sum_{j=1}^p (\xi''_{k,j}
-E\tau(\xi''_{k,j}))-(\eta''_{k,j}-E\tau(\eta''_{k,j}))\right|
\Rightarrow0
\endaligned \tag1.8
$$
hold where $\Rightarrow$ denotes stochastic convergence. The logarithm
of the characteristic function of the sum
$\summ_{j=1}^{n_k}(\eta''_{k,j}-E\tau(\eta''_{k,j}))$
can be expressed by means of a canonical measure $\bar M_k''$
close to the canonical measure $M_k''$ as
$$
\log E\exp\left\{it\(\sum_{j=1}^{n_k}
(\eta''_{k,j}-E\tau(\eta''_{k,j}))\)\right\}=\int
\frac{e^{itu}-1-it\tau(u)}{u^2}\bar M''_k(\,du), \tag1.9
$$
where $\bar M_k''(\,du)=M''_k(\,du)+\summ_{j=1}^{n_k}
\frac{\bar p_{k,j}-p_{k,j}}{p_{k,j}}G'_{k,j}(\,du)$, and $G'_{k,j}$
is the restriction of the measure $G_{k,j}$ to the set 
$\bold R^1\setminus [-\e_k,\e_k]$.

The triangular array $\eta'_{k,j}-E\eta'_{k,j}$, $k=1,2,\dots$, $1\le
j\le n_k$, satisfies the central limit theorem with a Gaussian limit
which has expectation zero and variance~$M_0(\{0\})$.} \medskip\noindent
 
We formulate a result about the convergence of infinitely divisible
distributions which enables us to complete the proof of Theorem~1.
Let us recall the following result discussed in Part~I of this work.
If~$M$ is a canonical measure on the real line, $\xi_n$,
$n=1,2,\dots$, is a Poisson process on the real line with counting
measure $\mu(du)=\frac{M(\,du)}{u^2}$, then the appropriate
regularized version of the sum $\eta=\eta_M=\summ_{n=1}^\infty \xi_n-
E\(\summ_{n=1}^\infty \tau(\xi_n)\)$ of the points of the Poisson
process, where $\tau(\cdot)$ is the function introduced in formula
(1.2) is convergent, and the random variable $\eta_M$ has infinitely
divisible distribution. More precisely, we can define the regularized
sum by the formula
$$
\eta_M=\limm_{L\to\infty}\(\(\summ_{n\colon\;|\xi_n|>2^{-L}}
\xi_n\)-E\(\sum_{n\colon\; |\xi_n|>2^{-L}}\tau(\xi_n)\)\).
$$
and this limit is convergent with probability one. The distribution
of the so defined random variable $\eta_M$ has such a characteristic
function whose logarithm equals
$$
\log\varphi(t)=\log\varphi_{M}(t)
=\int\frac{e^{itu}-1-it\tau(u)}{u^2}M(\,du). \tag1.10
$$
We shall call the random variable $\eta_M$ defined in such a way
the infinitely divisible random variable determined by the
Poisson process  $\xi_1,\xi_2,\dots$ with counting measure~$\mu$.
Now we formulate the following result.

\medskip\noindent
{\bf Proposition~2.} {\it Let $M_k$, $k=1,2,\dots$, be a sequence of
canonical measures which converges weakly to a canonical measure $M_0$.
Let us also assume \ that the relations $M_0(\{0\})=0$ and
$M_k(\{0\})=0$, $k=1,2,\dots$, hold. Put
$\mu_k(du)=\frac{M_k(du)}{u^2}$, $k=0,1,2,\dots$. Then we can define
Poisson processes $\xi_{k,1},\xi_{k,2},\dots$ with counting measure
$\mu_k$ and Poisson processes $\bar\xi_{k,1},\bar\xi_{k,2},\dots$ with
counting measure $\mu_0$ in such a way that the random variables
$\eta_k$  with infinitely divisible distribution determined by the
Poisson processes $\xi_{k,1},\xi_{k,2},\dots$ (introduced e.g. before
formula 1.10) and the random variables
$\bar\eta_k$ with infinitely divisible distribution determined by
the Poisson processes $\bar\xi_{k.1},\bar\xi_{k,2},\dots$ satisfy
the relation $\eta_k-\bar\eta_k\Rightarrow0$ where $\Rightarrow$
denotes stochastic convergence. (Let us remark that the distributions
of the random variables $\bar\eta_k$ do not depend on the index~$k$.)}
 
\medskip\noindent
{\it Remark:}\/ Proposition~2 and Theorem~A together imply that if the
measures $M_k$, $k=0,1,2,\dots$, satisfy the conditions of
Proposition~2, then the distributions of the random variables $\eta_k$
defined in Proposition~2 converge in distribution to the distribution
function whose characteristic function is given in formula~(1.3). We
also remark that with the help of some additional work a stronger
version of Proposition~2 could also be proved. It is possible to make
such a construction in which $\bar\eta_k=\bar\eta$, that is these
random variables (and the Poisson processes determing them) do not
depend on the index~$k$. Further, it can be achieved that also the
relation $\eta_k-\bar\eta_k\to0$ hold with probability~one. But for our
purposes the weaker result formulated in Proposition~2 is as good as its
above mentioned stronger version. \medskip\noindent
 
We show that Propositions~1 and~2 together with Theorem~A imply
Theorem~1. Let us consider the random variables $\eta_{k,j}'$,
$\eta_{k,j}''$, and $\eta_{k,j}=\eta_{k,j}'+\eta_{k,j}''$,
$k=1,2,\dots$, $1\le j\le n_k$, defined in the coupling construction
given after Lemma~1 together with the sums
$T_k=\summ_{j=1}^{n_k}(\eta_{k,j}-E\tau(\eta_{k,j}))$,
$T'_k=\summ_{j=1}^{n_k}(\eta_{k,j}'-E\eta_{k,j}')$ and
$T_k''=\summ_{j=1}^{n_k}(\eta_{k,j}''-E\tau(\eta_{k,j}''))$
defined with their help. First we claim that the sums $T_k$
converge in distribution to an infinitely divisible distribution
function whose characteristic function has a logarithm given by
formula~(1.3). Indeed, $T_k=T'_k+T_k''$, the random variables $T'_k$
and $T_k''$ are independent, and the random variables $T'_k$ converge
in distribution to the Gaussian distribution with expectation
zero and variance $M_0(\{0\})$ by Proposition~1. On the other hand,
the random variables $T_k''$ converge in distribution to an infinitely
divisible distribution determined by the canonical measure $M''_0$
because of Proposition~2, formula (1.9) and the convergence of the 
canonical measure $\bar M_k''$ to the canonical measure $M_0''$. (The 
measure $M_0''$ is the restriction of the measure $M_0$ to the set 
$\bold R^1\setminus\{0\}$.) The convergence of $\bar M_k''$ to $M''-0$
follows from the form of the measures $\bar M''_k$, the convergence of 
the measures $M''_k$ to $M''-0$ by Lemma~1 and the convergence of the
measures
$\summ_{j=1}^{n_k}\frac{\bar p_{k,j}-p_{k,j}}{p_{k,j}}G_{k,j}(\,du)$
to the measure identically zero on the real line. The last convergence 
holds, since $\frac{\bar p_{k,j}-p_{k,j}}{p_{k,j}}\ge0$, and
$\limm_{k\to\infty}\supp_{1\le j\le n_k}
\frac{\bar p_{k,j}-p_{k,j}}{p_{k,j}}\to0$. Indeed, by the identity
$1-e^{-\bar p_{k,j}}=p_{k,j}$ defining the quantity $\bar p_{k.j}$ we 
have $\bar p_{k,j}=-\log(1-p_{k,j})$. Hence we have 
$p_{k,j}\le\bar p_{k,j}\le p_{k,j}+p_{k,j}^2$ for large indices~$k$. 
(Here we exploit that for a large index~$k$ all numbers~$p_{k,j}$ 
are very small. Hence the relations formulated for the quantities
$p_{k,j}$ and $\bar p_{k,j}$ really hold.
 
Finally Proposition~1 and Theorem~A enable us to prove Theorem~1, i.e.\
the statement that the random sums $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$, or
what is equivalent to it, the random sums $\tilde
S_k=\summ_{j=1}^{n_k}\tilde \xi_{k,j} =\summ_{j=1}^{n_k}\(\tilde
\xi_{k,j}-E\tau(\tilde \xi_{k,j})\)$ of the random variables
$\tilde\xi_{k,j}$ defined in the coupling construction described after
Lemma~1  satisfy the statement of Theorem~1. Indeed, formula~(1.8)
implies that $\tilde S_k-T_k\Rightarrow0$ where $\Rightarrow$ denotes
stochastic convergence. Hence the random variables $S_k$ or $\tilde S_k$
converge to the same distribution function as the random
variables~$T_k$, and this implies Theorem~1.
\medskip
 
Let us make some comments about the proof of Theorem~1 explained in
this paper. This approach also may explain that although Theorem~1
supplies many cases when the normalized sums of independent random
variables have a limit distribution, the central limit theorem, i.e.\
the case when the limit is Gaussian deserves its name, the limit
theorems with a Gaussian limit really play a central role in the
theory of limit theorems for sums of independent random variables.
Let us observe that in the coupling construction applied in this proof
we approximated each term which contributes to the non-Gaussian part
of the limit individually by a random variable with an infinitely
divisible distribution. Then we showed the sum of the errors caused by
these approximations is negligible. This fact can be interpreted in
such a way that limit theorems with a non-Gaussian limit
must have a very special form, in a certain sense the distributions of
the terms in the sum must resemble to the limit distribution. The
picture in the case of the central limit theorem is quite different. In
this case, --- and this is one of the most remarkable facts in
probability theory --- the distribution of the individual terms in the
sum may be quite general, the distribution of the sum ``forgets" the
distribution of the individual terms. In such a case a term by term
approximation of the summands independently of each other, --- and this
was done in the coupling construction for the non-Gaussian part ---
would cause a non-negligible error.
 
The method of proof given here and in Part~II was quite different.
Here we applied the so-called coupling method and proved the result
by means of a probabilistic argument. In the proof of Part~II the
characteristic function technique, a useful method of analysis was
applied. Nevertheless, it may be useful to understand that these two
approaches are not so far from each other as it may seem at first sight.
The coupling argument is also present in a hidden way also in the proof
by means of characteristic functions.
 
Indeed, let us consider a triangular array
$\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le
n_k$ which satisfies the uniform smallness condition. Let
$\varphi_{k,j}(t)$ denote the characteristic function of the random
variable $\xi_{k,j}$. Put
$S_k=\summ_{j=1}^{n_k}\xi_{k,j}$. If the sums $S_k$ have a limit
distribution then the relation
$$
\lim_{k\to\infty}\prod_{j=1}^{n_k}\varphi_{k,j}(t)=\psi(t)
$$
holds with an appropriate characteristic function $\psi(t)$.
We have seen the (non-trivial) fact that in the last formula logarithm
can be taken, i.e.\ this relation is equivalent to the formula
$$
\lim_{k\to\infty}\summ_{j=1}^{n_k}\log\varphi_{k,j}(t)=\log\psi(t)
$$
Another important step of the proof was to show that since the random
variables $\xi_{k,j}$ are relatively small the replacement of the term
$\log\varphi_{k,j}(t)$ by $\varphi_{k,j}(t)-1$ is allowed, i.e. we
can write
$$
\lim_{k\to\infty}\summ_{j=1}^{n_k}(\varphi_{k,j}(t)-1)=\log\psi(t).
$$
Let us also observe that, as we have seen in Part~I, the function
$\varphi_{k,j}(t)-1$ is the logarithm of the characteristic function
of the infinitely divisible random variable which is determined by the
Poisson process whose counting measure is the distribution function
$F_{k,j}$ of the random variable $\xi_{k,j}$. In such a way the
replacement of the function $\log\varphi_{k,j}(t)$ by
$\varphi_{k,j}(t)-1$ corresponds to the coupling construction made in
this part of the work.
 
Finally, we remark that the above coupling method enables us to
approximate not only the sums $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ by means
of random variables with infinitely divisible distribution, but under
some natural conditions the partial sums
$S_{k,l}=\summ_{j=1}^{l}\xi_{k,j}$, $1\le k\le n_k$, can be approximated
simultaneously by means of sums of independent random variables with
indivisible distribution. This argument leads to the investigation of
the so-called functional limit theorems, a result which deserves a more
detailed discussion.
\medskip\noindent
{\script A.) Functional limit theorems for general triangular
arrays.} \medskip
 
An interesting and important result of probability theory, called the
functional central limit theorem or invariance principle in the
literature, states that if a triangular array satisfies the central
limit theorem, then the broken line processes made from the partial
sums of these random variables in a natural way converge weakly to a
Wiener process. More explicitly, let $\xi_{k,j}$, $k=1,2,\dots$, $1\le
j\le n_k$, be a triangular array such that $E\xi_{k,j}=0$,
$k=1,2,\dots$, $1\le j\le n_k$, $\limm_{k\to\infty}\summ_{j=1}^{n_k}
E\xi^2_{j,k}=1$, and the Lindeberg condition
$\limm_{k\to\infty}\summ_{j=1}^{n_k}E\xi_{j,k}^2I(|\xi_{j.k}|>\e)=0$
holds for all $\e>0$. Define the partial sums $S_{k,l}=\summ_{j=1}^l
\xi_{k,j}$, $k=1,2,\dots$, $1\le l\le n_k$, the numbers $u_{k,0}=0$
and $u_{k,l}=\frac1{D_k}\summ_{j=1}^l E\xi_{k,j}^2$, $1\le l\le n_k$,
in the interval $[0,1]$ where $D_{k}=\summ_{j=1}^{n_k} E\xi_{k,j}^2$.
Then we can define with the help of these quantities the random broken
lines $S_k(t)$, $0\le t\le1$, in such a way that $S_k(0)=0$,
$S_k(u_{k,l})=S_{k,l}$, $1\le l\le n_k$, and the functions $S_k(t)$
are linear in all intervals $[u_{k,l-1},u_{k,l}]$, $1\le l\le n_k$. The
functional central limit theorem states that the distributions of the
stochastic processes $S_k(t)$, $0\le t\le 1$, considered as
$C([0,1])$ valued random variables, converge weakly to the distribution
of a Wiener process. Let us emphasize that this result states in
particular that the necessary and sufficient condition of the central
limit theorem implies at the same time a stronger result.
 
The question arises whether the general limit theorems considered in
this work have a similar functional limit version. There is a positive
answer to this question. The limit theorem formulated in Theorem~1 holds
if and only if an appropriately defined sequence of canonical measures
on the real line converges to a canonical measure~$M_0$. We shall show
that we can introduce and define an appropriate sequence of canonical
measures on the strip $\bold R^1\times[0,1]$ which are closely related
to the canonical measures considered in Theorem~1, and their convergence
implies a functional limit theorem for the appropriately defined random
broken lines made from the sums of partial sums of the random variables
in the triangular array. We shall formulate such a result in Theorem~2.
But to do this first we have to introduce some definitions and
notations.
 
To formulate Theorem~2 first we introduce the appropriate function
space where we shall work. This space is called the $D([0,1])$
space in the literature.
 
We say that a function $x(t)$, $0\le t\le1$, is a cadlag function
(continue \`a droite, limite \` a gauche) if the function $x(t)$ is
continuous from the right, and it also has a left-hand
side limit in all points. The space $D([0,1])$ consists of the cadlag
functions in the interval~$[0,1]$, and an appropriate distance is
introduced in it. A possible definition of this metric is the distance
$d(\cdot,\cdot)$ defined in the following way: Let $x,y\in D([0,1])$ be
two cadlag functions and $\e>0$  a real number. The relation $d(x,y)\le
\e$ holds if there exists a strictly monotone function $\lambda(t)$
which is a homeomorphism of the interval $[0,1]$ into itself,
$\supp_{0\le t\le1}|\lambda(t)-t|\le\e$, and $\supp_{0\le t\le
1}|y(t)-x(\lambda(t))| \le\e$.
 
The space $D([0,1])$ is a separable metric space with the above
distance, but it is not a complete metric space. The property that two
cadlag functions $x(\cdot)$ and $y(\cdot)$ are close to each other with
respect to the metric $d(\cdot,\cdot)$ means that although these two
functions may be far from each other with respect to the supremum norm,
but they can put close to each other with respect to the this norm
if the argument of one of these functions is slightly perturbed in an
appropriate way. We introduced the space $D([0,1])$ because we need this
notion in the formulation of Theorem~2. Here we only formulate the
results about this space but omit the proofs. All of them can be found
in P.~Billingsley's book ``Convergence of probability measures". We had
to introduce this notion, because the possible limit processes in
Theorem~2, --- the Poisson process is a typical example, --- do not
have continuous trajectories, hence we have to work in a different
function space.

Let us remark that the above metric is not the only possible good
metric which can be introduced in the space $D([0,1])$. For instance
the following metric $d_0(\cdot,\cdot)$ in the space $D([0,1])$ is
often applied in the literature. Let $x(\cdot)$, $y(\cdot)$ be two
cadlag functions. We say that $d_0(x,y)\le\e$ if there exists such a
homeomorphism $\lambda(\cdot)\colon\;[0,1]\to [0,1]$ of the interval 
$[0,1]$ into itself for which $\lambda(0)=0$, $\supp_{t\neq s}\log
\left|\frac{\lambda(t)-\lambda(s)}{t-s}\right|\le\e$, and
$|x(t)-y(\lambda(t))|\le\e$ for all numbers $t\in [0,1]$. It can be
proved that the metrics $d(\cdot,\cdot)$ and $d_0(\cdot,\cdot)$ define
the same topology on the space $D([0,1])$. This means that sequences of
probability measures on the space $D([0,1])$ simultaneously converge or
do not converge with respect to these metrics. Hence they are
equivalent for our purposes. The essential difference between these two
metrics is that the space $D([0,1])$ is a separable {\it complete}\/
metric space with respect to the metric $d_0(\cdot,\cdot)$, but this
relation does not hold for the metric $d_0(\cdot,\cdot)$. Some proofs
are simpler in complete metric space, and this is the main reason why
the metric $d_0(\cdot,\cdot)$ is applied in several cases.
 
Let us remark that in the formulation of Theorem~A we only assumed
that the metric space $(X,\rho)$ we have considered is separable, but
did not demand that it has to be complete. This makes possible to
apply the metric $d(\cdot,\cdot)$ in subsequent proofs, and this
simplifies certain arguments.
 
Now we define the stochastic processes and canonical measures which
will appear in Theorem~2. Let a triangular array $\xi_{k,j}$, $1\le
j\le n_k$, be given, and let $F_{k,j}$ denote the distribution function
of the random variable~$\xi_{k,j}$. Let us define the partial sums
$$
S_{k,0}=0, \quad S_{k,l}=\sum_{j=1}^l \xi_{k,j},\quad 1\le l\le n_k
\tag1.11
$$
Then let us fix for all numbers $k=1,2,\dots$  an appropriate sequence
of numbers $0=u_{k,0}\le u_{k,1}\le u_{k,2}\le\cdots\le u_{k,n_k}=1$
and define the random cadlag functions in the interval~$[0,1]$ as
$$
\aligned
S_{k}(t)=S_k(t,u_{k,0},\dots,u_{k,n_k})&=S_{k,l-1},
\quad\text{if } u_{k,l-1}\le t< u_{k,l}, \;\; 1\le l\le n_k, \\
S_k(1)&=S_{k,n_k},
\endaligned \tag1.12
$$
$k=1,2,\dots$, with their help. (These numbers $0=u_{k,0}\le u_{k,1}\le
u_{k,2}\le\cdots\le u_{k,n_k}=1$ are needed to define the appropriate
scaling in the definition of the random cadlag functions. We cannot
give them in such an explicit way as in the functional central limit
theorem.) Furthermore, let us define certain $\sigma$-finite measures
$N_k$ on the direct product of the real line and the interval
$[0,1]$, on the set $\bold R^1\times [0,1]$, in the following way:
Let $0=u_{k,0}\le u_{k,1}\le u_{k,2}\le \cdots\le u_{k,n_k}=1$ be the
same sequence of numbers which appeared in formula~(1.12).
$$
\text {The measure } N_k(\cdot)\text{ is concentrated on the set }\bold
R^1\times \bigcupp_{l=1}^
{n_k}\{u_{k,l}\}, \tag1.13
$$
and the restriction of the measures $N_k(\cdot)$ to the lines
$\{(t,u)\colon\;t\in \bold R^1, u=u_{k,l}\}$, $1\le l\le n_k$, equals the
measure $x^2F_{k,l}(dx)$, i.e.
$$
N_k(B\times\{u_{k,l}\})=\int_B x^2 F_{k,l}(\,dx),\quad \text{if } 1\le
l\le n_k, \text{ and } B\subset \bold R^1 \text{ is a measurable
set.} \tag1.14
$$
 
Let us define the notion of canonical measures on the strip
$\bold R^1\times [0,1]$ and their convergence. The measures $N_k$
defined in formulas (1.13) and (1.14) also satisfy the properties of
canonical measures.  \medskip\noindent
{\bf The definition of canonical measures and their convergence.} {\it
We call a $\sigma$-finite measure $N(\cdot)$ on the strip $\bold
R^1\times [0,1]$ canonical if for all numbers
$s>0$
$$
N([-s,s]\times [0,1])<\infty,\quad\text{and}\quad
\int_{\{(u,v)\colon\; |u|>s,\; 0\le v\le1\}}\frac{N(\,du,\,dv)}{u^2}<\infty.
$$
Let a sequence $N_k$, $k=0,1,2,\dots$, of canonical measures be given
on the strip $\bold R^1\times [0,1]$. We say that this sequence of
canonical measures $N_k$ converges (weakly) to a canonical measure
$N_0$ on the strip $\bold R^1\times [0,1]$ as $k\to\infty$ if for all
such numbers $0\le a\le b\le1$ which are points of continuity of the
limit measure $N_0$, i.e. for which
$\limm_{\e\to0}N_0([-R,R]\times[a-\e,a+\e])=0$,
$\limm_{\e\to0}N_0([-R,R]\times[b-\e,b+\e])=0$ and
$\limm_{\e\to0}\int_{|u|>R,\,|v-a|<\e} \frac{N_0(du,dv)}{u^2}=0$,
$\limm_{\e\to0}\int_{|u|>R,\,|v-b|<\e} \frac{N_0(du,dv)}{u^2}=0$
for all numbers $R>0$, the canonical measures $M_{k,a,b}$,
$k=1,2,\dots$, on the real line defined by the formula
$M_{k,a,b}(B)=N_k(B\times[a,b])$ for all measurable sets $B\in\bold
R^1$ converge to the canonical measure $M_{0,a,b}(B)$ on the real
line, where $M_{0,a,b}(B)=N_0(B\times[a,b])$, for all measurable sets
$B\in\bold R^1$.}\medskip
 
Now we formulate Theorem~2. It says that if the canonical measures
$N_{k}$ on the strip $\bold R^1\times [0,1]$ defined in formulas
(1.13) and (1.14) converge to a canonical measure $N_0$ on
the strip $\bold R^1\times [0,1]$ then the stochastic processes
$S_k(\cdot)$ defined in formula (1.12), considered as $D([0,1])$
space valued random variables, converge weakly to a random process
with cadlag trajectories which is determined by the limit
canonical measure $N_0$ in a natural way.
\medskip\noindent
{\bf Theorem~2.} {\it Let $\xi_{k,j}$, $k=1,2,\dots$, $1\le
j\le n_k$, be a triangular array satisfying the uniform smallness
condition and such that $E\tau(\xi_{k,j})=0$ far all $k=1,2,\dots$,
$1\le j\le n_k$ with the function $\tau(x)=\tau_a(x)$ defined in
formula (1.2) where $a>0$ is some fixed number. Let us fix for all
numbers $k=1,2,\dots$ a sequence of numbers $0=u_{k,0}\le u_{k,1}\le
u_{k,2}\le\cdots\le u_{k,n_k}=1$ satisfying the relation
$\limm_{k\to\infty}\supp_{1\le l\le n_k}|u_{k,l}-u_{k,l-1}|=0$, and
let us consider the canonical measures $M_k$ defined on the strip
$\bold R^1\times [0,1]$ defined by formulas (1.13) and (1.14) with the
above sequences of numbers $0=u_{k,0}\le u_{k,1}\le u_{k,2}\le
\cdots\le u_{k,n_k}=1$ and the distribution functions $F_{k,l}$ of the
random variables $\xi_{k,l}$, $k=1,2,\dots$, $1\le l\le n_k$. Let us
assume that these canonical measures $N_k$ on the strip $\bold R^1\times
[0,1]$ converge weakly to a canonical measure $N_0$ which satisfies the
relation $N_0(\bold R^1\times\{0\})=0$. Furthermore, let us also assume
that 

\medskip
\item{a.)} The function $\lambda(t)=N_0(\{0\}\times [0,t])$ defined
with the help of the canonical measure $N_0$ is continuous in the
interval $[0,1]$.
\item{b.)} For all numbers $b>0$ the function
$$
\nu_b(t)=\int_{\{(x,y)\colon\; |x|>b, \; 0\le y\le
t\}}\frac {N_0(dx, dy)}{x^2},\quad 0\le t\le 1,
$$
defined with the help of the limit canonical measure $N_0$
is continuous in the interval $[0,1]$.

\medskip
Then the stochastic processes $S_k(t)$, $0\le t\le1$, defined in
formula (1.12) considered as $D([0,1])$ valued random variables
converge weakly to a stochastic process $S(t)$, $0\le t\le1$, which
can be considered as a $D([0,1])$ valued random variable. This
process $S(t)$ is a stochastic process with cadlag
trajectories, it has independent increments, and it is determined
by a Poisson field on $\bold R^1\times[0,1]$ with counting measure
$\frac{N_0(du,dv)}{u^2}$ in a natural way. The distribution of this
stochastic process can be described in the following way. We have
$S_0\equiv0$, and since $S(t)$ is a process with independent
increments, it is enough to give the distribution of the increments
$S(v)-S(u)$, $0\le u\le v\le1$. The distribution function of such a
difference is an infinitely divisible distribution determined by the
canonical measure $\frac{M_{0,u,v}(dx)}{x^2}$ with $M_{0,u,v}(dx)=
N_0(dx\times(u,v])$. The characteristic function $\varphi_{u,v}(\cdot)$
of the random variable $S(v)-S(u)$ has a logarithm which is given by
the following modified version of formula (1.3):
$$
\log \varphi_{u,v}(t)=\int_{-\infty}^\infty
\frac{e^{itx}-1-it\tau(x)}{x^2} M_{0,u,v}(\,dx). \tag1.15
$$
}

\medskip\noindent
{\it Remark 1.} We showed in Lemma~2 in Section~5 of Part~I that a
stochastic process with the properties demanded for the limit
stochastic process appearing in Theorem~2 really exists. The Poisson
process with counting measure $\frac{N_0(du,dv)}{u^2}$ defines with
the help of appropriate regularized sums investigated in Part~I a
stochastic process $S(t)$, $0\le t\le1$, with the required properties.
In our discussion we sometimes regard a stochastic process with cadlag
trajectories as a random variables $D([0,1])$. This is legitimate, but
to justify our right to do this we have to solve a non-trivial measure
theoretical problem. We have to show that the measurability of a
stochastic process with cadlag trajectories in the usual way is
equivalent to its measurability as a function in the space $D([0,1])$
with respect to the $\sigma$-algebra determined by the topology of the
space $D([0,1])$. The proof of this result can be found for instance in
Billingsley's book {\it Convergence of Probability Measures}\/
Theorem~14.5. 

\medskip\noindent
{\it Remark~2.} The conditions a.) and b.) imposed for the limit
canonical measure $N_0$ can be formulated in a unified way as
$N_0(\bold R^1\times\{t\})=0$ for all $0\le t\le1$. This condition
cannot be dropped. We show this with the help of an example
explained in a rather sketchy way. Let us consider a triangular array
$\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le k$, (i.e. $n_k=k$) which is
defined in the following way. Consider a sequence of numbers $\e_k$,
$k=1,2,\dots$, such that $\limm_{k\to\infty}\e_k=0$ and
$\limm_{k\to\infty}k\e_k=\infty$. Let $\xi_{k,j}=\zeta_{k,j}
+\eta_{k,j}$ if $(\frac12-2\e_k)k<j\le (\frac12-\e_k)k$ or
$(\frac12-4\e_k)k<j\le (\frac12-3\e_k)k$ and $\xi_{k,j}=\zeta_{k,j}$
if $j\in [1,k]\setminus(((\frac12-2\e_k)k,\frac12-\e_k)k]
\cup((\frac12-2\e_k)k, \frac12-\e_k)k])$ where the $\zeta_{k,j}$ are
independent Gaussian random variables with expectation zero and
variance $\frac1k$, and $\eta_{k,j}$ are independent random
variables, $P(\eta_{k,j}=1)=1-P(\eta_{k,j}=0)
=\frac{\e_k}{2k}$ which are also independent of the random
variables $\zeta_{k,j}$. Define the numbers $u_{k,l}$ appearing in
the formulation of Theorem~2 by the formula $u_{k,l}=\frac lk$, $0\le
l\le k$. (Actually, it was not necessary to take a Gaussian part
$\zeta_{k,j}$ in this example, we only introduced it to make the
choice of the numbers $u_{k,l}$ more natural.) It is not difficult to
see that in this case the canonical measures $N_k$ introduced before
the formulation of Theorem~2 converge to the canonical measure
$N_0=N_0'+N_0''$, where $N_0'$ is the Lebesgue measure on the
set $\{0\}\times[0,1]$, and the measure $N''_0$ is concentrated in the
point $(1,\frac12)$, $N_0''(\{(1,\frac12)\})=1$. We show that, as the
limit measure $N_0$ does not satisfies condition b.), the processes
$S_k(t)$ do not converge to the processes determined by the measure
$N_0$. To see this observe that if we disregard the Gaussian part of
this example, i.e.\ we consider the partial sums
$S''_{k,l}=\summ_{j=1}^l\eta_{k,j}$, $1\le l\le k$, the stochastic
processes $S''_k(t)=S_{k,l}$ if $u_{k,l}\le t<t_k$, and the candidate
for the limit process is the process determined by the measure $N_0''$.
This limit process $S_0(t)$, $0\le t\le1$, is defined as $S_0(t)=0$
if $0\le t<\frac12$ and $S_0(t)=\eta$ if $\frac12\le t\le1$ where
$\eta$ is a Poissonian random variable with parameter~1. The
distributions of the processes $S_k(\cdot)$ do not converge to the
distribution of the process $S_0(\cdot)$. Indeed, the random variables
$S_k(\frac12-4\e_k)$, $S_k(\frac12-3\e_k)$, $S_k(\frac12-\e_k)$ take
three different integer values with a positive probability, while the
process $S''_0(\cdot)$ can take at most two values with probability one.
This excludes the convergence of the distributions of the processes
$S_k(\cdot)$ to those of the process $S_0(\cdot)$ in the $D([0,1])$
space. Since the Gaussian part of the processes $S_k(\cdot)$ have
very small fluctuation in the interval $[\frac12-4\e_k,\frac12]$ the
original example also yields a counter-example.  \medskip
 
The proof of Theorem~2 will be similar to that of Theorem~1. We can
split the stochastic processes $S_k(\cdot)$ with the help of
Proposition~1 and Lemma~1 to two parts, one of them responsible for the
Gaussian the other one for the Poissonian part of the limit. The
convergence to the Gaussian part of the limit follows from the
functional central limit theorem. Then  the converge to the
Poissonian part of the limit process can be investigated by a
refined version of the coupling argument applied in the proof of
Theorem~1. Let us remark that the expression $\supp_{1\le j\le n_k}$
appeared in formula~(1.8) because it was appropriate in this form in
the proof of Theorem~2.
 
\beginsection 2. The proof of Theorem 1.
 
In this section we prove the results applied in the proof of Theorem~1,
Lemma~1 and Propositions~1 and~2.
 
\medskip\noindent
{\it The proof of Lemma~1.}\/ Let us choose a monotone decreasing
sequence of positive numbers $\eta_p$, $p=1,2,\dots$, such that
$\limm_{p\to\infty}\eta_p=0$ and the numbers $\pm\eta_p$ are points of
continuity of the measure $M_0$. Then 
$\limm_{p\to\infty}M_0((-\eta_p,\eta_p))=M_0(\{0\})$. Beside this, 
there exists a threshold index $k_0(p)$ for all numbers $p=1,2,\dots$
such that for all $k\ge k_0(p)$ the inequality 
$|M_k((-\eta_p,\eta_p))-M_0(-\eta_p,\eta_p))|\le\frac1p$ holds, and 
if the origin is a point of continuity of the
measure $M_0$, i.e.\ $M_0(\{0\})=0$, then also the inequality
$|M_k((0,\eta_p))-M_0(0,\eta_p))|\le\frac1p$ holds. We may also assume
that $|M_0^\pm(\eta_p)-M^\pm_k(\eta_p)|\le\frac1p$ if $k\ge k_0(p)$
where $M_0^\pm(\cdot)$ and the functions $M_k^\pm(\cdot)$ are the
functions defined in formula (1.4). Because of the uniform smallness
condition we can guarantee that
$$
\supp_{1\le j\le n_k}(1-F_{k,j}(\eta_p))+\supp_{1\le j\le n_k}
F_{k,j}(-\eta_p) \le\frac1{p(M_0^+(\eta_p)+M_0^-(\eta_p))+2}
$$
if $k\ge k_0(p)$, and the threshold index $k_0(p)$ is chosen
sufficiently large. We may also assume that the sequence of threshold
indices $k_0(p)$, $p=1,2,\dots$, is monotone increasing. Put
$\e_k=\eta_p$ if $k_0(p)\le k<k_0(p+1)$. With such a choice of the
numbers $\e_k$ the statements of Lemma~1 hold. Indeed,
$$
\limm_{k\to\infty}M'_k([a,b])=\limm_{k\to\infty} M'_k([-\eta_{k_0(p)},
\eta_{k_0(p)}])=M_0(\{0\})=M_0'([a,b])
$$
if the interval $[a,b]$ contains the origin in its interior, and
$\limm_{k\to\infty}M'_k([a,b])=0=M_0'([a,b])$ if the interval $[a,b]$
does not contain the origin in its interior and the points $a$  and $b$
are points of continuity of the measure $M_0$ (in particular also in
the case if 0 is a point of continuity of the measure $M_0$, and
$a=0$ or $b=0$). These relations together with the fact that the
measures $M'_k$, $k=1,2,\dots$, and $M_0'$ are concentrated in a finite
interval $[-A,A]$ imply that the measures $M'_k$ weakly converge to
the measure $M_0'$. As the sequence of measures $M_k$  converges
(weakly) to the measure $M_0$ and the sequence of measures $M'_k$
converges (weakly) to the measure $M_0'$ also the measures
$M''_k=M_k-M'_k$ converge (weakly) to the measure $M_0''=M_0-M_0'$.
Furthermore,
$$
\align
&\sum_{j=1}^{n_k}[(1-F_{k,j}(\e_k))+F_{k,j}(-\e_k)]^2\\
&\qquad\le \sup_{1\le j\le n_k}[(1-F_{k,j}(\e_k))+F_{k,j}(-\e_k)]
\sum_{j=1}^{n_k}[(1-F_{k,j}(\e_k))+F_{k,j}(-\e_k)]\\
&\qquad=\sup_{1\le j\le
n_k}[(1-F_{k,j}(\e_k))+F_{k,j}(-\e_k)](M^+_k(\e_k)+M_k^-(\e_k))\\
&\qquad\le\frac1{p(M_0^+(\eta_p)+M_0^-(\eta_p))+2}
\(M_0^+(\e_k)+M_0^-(\e_k)+\frac2p\)\le\frac1p
\endalign
$$
if $k\ge k_0(p)$. From here we get formula~(1.5) by
 taking the limit procedure $k\to\infty$.
\medskip\noindent
{\it Proof of Proposition~1.}\/ As
$\tilde \xi_{k,j}=\eta'_{k,j}I(\zeta_{k,j}=0)+\gamma_{k,j,1}I
(\zeta_{k,j}\ge1)$, and the random variables $\zeta_{k,j}$ are
independent of the other random variables, hence
$$
\align
P(\tilde\xi_{k,j}\in A)&=
P(\tilde\xi_{k,j}\in A|\zeta_{k,j}=0)P(\zeta_{k,j}=0)+
P(\tilde\xi_{k,j}\in A|\zeta_{k,j}\ge1)P(\zeta_{k,j}\ge1)\\
&=P(\eta'_{k,j}\in A)P(\zeta_{k,j}=0)+
P(\gamma_{k,j,1}\in A)P(\zeta_{k,j}\ge1)\\
&=\bar\nu(A)(1-p_{k,j})+\bar{\bar\nu}(A)p_{k,j}\\
&=P(\xi_{k,j}\in A\cap\{x\colon\; |x|< \e_k\})+
P(\xi_{k,j}\in A\cap\{x\colon\; |x|\ge \e_k\})\\
&=P(\xi_{k,j}\in A),
\endalign
$$
for all measurable sets $A\subset \bold R^1$, that is the random
variables $\tilde\xi_{k,j}$ and $\xi_{k,j}$ have the same distribution.
The coupling construction also implies that for a fixed index~$k$ the
random variables $\tilde\xi_{k,j}$, $1\le j\le n_k$, are independent.
This construction also implies that  for a fixed $k$ the random
variables $\eta'_{k,j}$, $\eta''_{k,j}$, $1\le j\le n_k$, are
independent, and
$P(|\xi'_{k,j}|\le\e_k)=P(|\eta'_{k,j}|\le\e_k)=1$. To prove the
relation
$\limm_{k\to\infty}\summ_{j=1}^{n_k}|E\xi'_{k,j}-E\eta'_{k,j}|=0$
let us observe that
$$
\align
|E\xi'_{k,j}-E\eta'_{k,j}|&=(1-P(\zeta_{k,j}=0))|E\eta'_{k,j}|
=\frac{1-P(\zeta_{k,j}=0)}{P(\zeta_{k,j}=0)}|E\xi'_{k,j}|\\
&=\frac{P(|\xi_{k,j}|\ge\e_k)}{P(|\xi_{k,j}|<\e_k)}
|E\tau(\xi''_{k,j})|\le 2P(|\xi_{k,j}|\ge\e_k)|E\tau(\xi''_{k,j})|
\endalign
$$
if $k\ge k_0$ with an appropriate constant~$k_0$, since
$0=E\tau(\tilde\xi_{k,j})=E\xi'_{k,j}+E\tau(\xi''_{k,j})$, and
$P(\zeta_{k,j}\ge 1)=1-e^{-\bar p_{k,j}}=p_{k,j}=P(|\xi_{k,j}|\ge
\e_k)$, where $\bar p_{k,j}$ is the solution of the equation
$p_{k,j}=1-e^{-\bar p_{k,j}}$, $p_{k,j}=P(|\xi_{k,j}|\ge\e_k)$, and
$p_{k,j}\le\frac12$ if $k\ge k_0$. As $|E\tau(\xi''_{k,j})|\le
aP(|\xi_{k,j}|\ge\e_k)$ this implies that
$|E\xi'_{k,j}-E\eta'_{k,j}|\le 2a P(|\xi_{k,j}|\le \e_k)^2
=2a\[(1-F_{k,j}(\e_k))+F_{k,j}(-\e_k)\]^2$. We get relation~(1.7) by
summing up these inequalities and applying formula~(1.5).
 
We can prove the first relation of formula~(1.8) with the help
of the Kolmogorov inequality. Indeed, for all numbers~$\e>0$
$$
\align
P&\(\sup_{1\le p\le n_k}\left|\sum_{j=1}^p(\xi'_{k,j}-E\xi'_{k,j})
-(\eta'_{k,j}-E\eta'_{k,j})\right|>\e\)
\le\frac{\summ_{j=1}^{n_k}\text{Var}\,(\xi'_{k,j}-\eta'_{k,j})}{\e^2}\\
&\qquad=\frac{\summ_{j=1}^{n_k}(1-P(\zeta_{k,j}=0))^2
\text{Var}\,\eta'_{k,j}}{\e^2}\le\frac1{\e^2}
\sup_{1\le j\le n_k}(1-P(\zeta_{k,j}=0))^2
\sum_{j=1}^{n_k}E{\eta'_{k,j}}^2\\
&\qquad=\frac1{\e^2}M_k([-\e_k,\e_k])
\sup_{1\le j\le n_k}P(|\xi_{k,j}|\ge\e_k)^2\to0\quad\text{if
}k\to\infty,
\endalign
$$
because $\limsupp_{k\to\infty}M_k([-\e_k,\e_k])\le
M_0(\{0\})<\infty$ in the construction of Lemma~1, and
$\supp_{1\le j\le n_k}(P(|\xi_{k,j}|\ge\e_k)^2\to0$ if $k\to\infty$.
 
To prove the second statement of formula~(1.8) first we show that
$$
E\tau(\eta''_{k,j})=\frac{\bar p_{k,j}}{p_{k,j}}
\int_{\{|u|\ge\e_k\}}\tau(u)F_{k,j}(\,du)=\frac{\bar p_{k,j}}{p_{k,j}}
E\tau(\xi_{k,j}''), \tag2.1
$$
where $p_{k,j}=P(|\xi_{k,j}|\ge\e_k)=[(1-F_{k,j}(\e_k)+F_{k,j}(-\e_k)]$,
and $\bar p_{k,j}$ is the solution of the equation 
$1-e^{-\bar p_{k,j}}=p_{k,j}$.
Indeed, by exploiting that if $\eta_1,\eta_2,\dots$, are independent,
identically distributed random variables, $\tau$ is a random variable
taking non-negative integer values which is independent of the random
variables $\eta_j$, then $E(\eta_1+\cdots+\eta_\tau)=E\tau E\eta_1$.
Further, since $E\zeta_{k,j}=\bar p_{k,j}=1-e^{-p_{k,j}}$ we get that
$$
\align
E\(\summ_{l=1}^{\zeta_{k,j}}\gamma_{k,j,l}I(|\gamma_{k,j,l}| \le a)\)
&=\frac {E\zeta_{k,j}}{p_{k,j}}\int_{\{\e_k\le|u|\le
a\}}\tau(u)F_{k,j}(\,du)\\
&=\frac{\bar p_{k,j}}{p_{k,j}} \int_{\{\e_k\le|u|\le
a\}}\tau(u)F_{k,j}(\,du),
\endalign
$$
and similarly
$$
\align
E\(\summ_{l=1}^{\zeta_{k,j}}I(|\gamma_{k,j,l}|\ge a)\)
&=\bar{\bar\nu}_{k,j}((-\infty,-a]\cup[a,\infty))E\zeta_{k,j}\\
&=\frac{\bar p_{k,j}}{p_{k,j}} \[1-F_{k,j}(a)+F(-a_{k,j})\].
\endalign
$$
Since
$$
E\tau(\eta''_{k,j})=E\(\summ_{l=1}^{\zeta_{k,j}}
\gamma_{k,j,l}I(|\gamma_{k,j,l}|\le a)\)+
aE\(\summ_{l=1}^{\zeta_{k,j}}I(|\gamma_{k,j,l}|\ge a)\),
$$
the above two identities imply formula~(2.1).
 
To prove the second relation of formula (1.8) let us also observe that
$$
\xi''_{k,j}-\eta''_{k,j}=\xi''_{k,j}-\sum_{l=1}^{\zeta_{k,j}}
\gamma_{k,j,l}= I(\zeta_{k,j}\ge2)\sum_{l=2}^{\zeta_{k,j}}
\gamma_{k,j,l},
$$
because by the coupling construction $\xi''_{k,j}=\eta''_{k,j}=0$, on
the set $\{\oo\colon\; \zeta_{k,j}(\oo)=0\}$ and
$\xi''_{k,j}=\eta''_{k,j}=\gamma_{k,j,1}$ on the set
$\{\oo\colon\;\zeta_{k,j}(\oo)\ge1\}$.  With the help of these 
relations we get that
$$
\align
P&\(\sum_{j=1}^p\xi''_{k,j}-\sum_{j=1}^p\eta''_{k,j}\neq
0\text{ for some number }1\le p\le n_k\)
\le\sum_{j=1}^{n_k}P(\zeta_{k,j}\ge2)\\
&\qquad\le \sum_{j=1}^{n_k}\[(1-F_{k,j}(\e_k)+F_{k,j}(-\e_k))\]^2\to
0\quad \text{if }k\to\infty
\endalign
$$
by the formula (1.5). Hence to prove the second relation of
formula~(1.8) it is enough to show that
$$
\lim_{k \to\infty}\sum_{j=1}^{n_k}|E\tau(\xi''_{k,j})
-E\tau(\eta''_{k,j})|=0.
$$
But by formula (2.1)
$$
|E\tau(\xi''_{k,j})-E\tau(\eta''_{k,j})|=
\left|\frac{\bar p_{k,j}-p_{k,j}}{p_{k,j}}
E\tau(\xi''_{k,j})\right| \le 2a p^2_{k,j}=2aP^2(|\xi_{k,j}|\ge\e_k)
$$
if $k\ge k_0$. (Observe that $|1-e^{-p_{k,j}}-p_{k,j}|<p_{k,j}^2$ and
$|E\tau(\xi''_{k,j}|\le 2ap_{k,j}$.) This implies that
$$
\sum_{j=1}^{n_k}|E\tau(\xi''_{k,j})
-E\tau(\eta''_{k,j})|\le 2a\summ_{j=1}^{n_k}
\[(1-F_{k,j}(\e_k)+F_{k,j}(-\e_k)\]^2\to0\quad \text{if } k\to\infty
$$
by formula (1.5). The above relations imply also the second part of
formula (1.8).
 
Finally, we get by summing up the identities in  formula (1.6) for the
numbers $1\le j\le n_k$   for a fixed integer $k$ and by applying the
definition given in Lemma~1 that
$$
\log E\exp\left\{it\(\sum_{j=1}^{n_k} (\eta''_{k,j}\)\right\}
=\int \frac{e^{itu}-1}{u^2}\bar M''_k(\,du),
$$
and athe summation of the identity (2.1) in the variable $j$ yields
the identity 
$$
\summ_{j=1}^{n_k} E\tau(\eta''_{k,j}))= \int
\frac{\tau(u)}{u^2}\bar M''_k(\,du).
$$ 
These relations imply formula (1.9).
The random variables $\eta_{k,j}'-E\eta_{k,j}'$, $1\le j\le n_k$,
are independent, $|\eta_k'-E\eta_k'|\le \e_k$, hence the sums
$T'_k=\summ_{j=1}^{n_k}(\eta'_{k,j}-E\eta_{k,j})$ satisfy the
central limit theorem. Beside this, $ET'_k=0$, and we complete the
proof of Lemma~1 if we show that
$\limm_{k\to\infty}\summ_{j=1}^{n_k}\text{Var}\,\eta_{k,j}'
=M_0(\{0\})$. This follows from the identity
$$
\lim_{k\to\infty}\sum_{j=1}^{n_k}\(E\eta_{k,j}'\)^2=0, \quad
\lim_{k\to\infty}\sum_{j=1}^{n_k}\left|{E\eta'_{k,j}}^2-{E\xi'_{k,j}}^2
\right|=0 \tag2.2
$$
to be proved below, since
$\limm_{k\to\infty}\summ_{j=1}^{n_k}{E\xi'_{k,j}}^2=M_0(\{0\})$.
 
We get similarly to the proof of formula (1.7) that
$$
\left|E\eta'_{k,j}\right|=\frac{|E\xi'_{k,j}|}{P(\zeta_{k,j}=0)}
=\frac{|E\tau(\xi''_{k,j})|}{P(|\xi_{k,j}|<\e_k)}
\le 2|E\tau(\xi''_{k,j})|\le 2aP(|\xi_{k,j}|>\e_k)
$$
if $k\ge k_0$ with an appropriate constant $k_0$. Then formula (1.5)
implies the first statement of formula~(2.2). On the other hand,
$$
\align
\left|E{\xi'_{k,j}}^2-E{\eta'_{k,j}}^2\right|&=
(1-P(\zeta_{k,j}=0))E{\eta'_{k,j}}^2
=\frac{1-P(\zeta_{k,j}=0)}{P(\zeta_{k,j}=0)}E{\xi'_{k,j}}^2\\
& \le 2P(|\xi_{k,j}|\ge\e_k)E{\xi'_{k,j}}^2.
\endalign
$$
This inequality together with the Schwarz inequality imply that
$$
\sum_{j=1}^{n_k}\left|E{\xi'_{k,j}}^2-E{\eta'_{k,j}}^2\right|\le
\(\sum_{j=1}^{n_k} 4P^2(|\xi_{k,j}|\ge\e_k)\cdot\sum_{j=1}^{n_k}
\(E{\xi'_{k,j}}^2\)^2\)^{1/2}\to0, \quad\text{if }k\to \infty,
$$
since $\limm_{k\to\infty}\summ_{j=1}^{n_k} P^2(|\xi_{k,j}|\ge\e_k)=0$ by
formula (1.5), and
$$
\summ_{j=1}^{n_k} \(E{\xi'_{k,j}}^2\)^2\le\const
\summ_{j=1}^{n_k} E{\xi'_{k,j}}^2\le \const
$$
with an appropriate constant for all numbers $k\ge1$.
 
\medskip\noindent
{\it The proof of Proposition~2.}\/ As the canonical measures $M_k$
converge to the canonical measure $M_0$ and $M_0(\{0\})=0$ for all
numbers $\e>0$ there exist such numbers $\delta=\delta(\e)>0$,
$R=R(\e)$ and threshold index $\bar n=\bar n(\e)$ for which
$$
M_k((-\delta,\delta))<\e^3,\quad \text{and}\quad
\int_{\{u\colon\; |u|>R\}}\frac1{u^2}M_k(\,du)<\e \quad \text{if } k\ge
\bar n(\e). \tag2.3
$$
We may also assume that the numbers $\pm\delta=\pm\delta(\e)$ and $\pm
R=\pm R(\e)$ are points of continuity of the measure $M_0$.
 
Let us introduce the measures $\mu_k(\,dx)=\frac{M_k(dx)}{x^2}$,
$k=1,2,\dots$ and $\mu_0(\,dx)=\frac{M_0(dx)}{x^2}$ on the real line.
Let us choose such numbers $\delta=x_1<x_2<\cdots<x_s=R$ with an
appropriate index $s$ for which $\pm x_l$ are points of continuity of
the measure $M_0$, $1\le l\le s$, and $\frac{\e^4}{2L}<x_l-x_{l-1}
<\frac{\e^4}L$, $1<l\le s$ with $L=\supp_{k\ge0}\mu_k
\((-R,-\delta)\cup(\delta,R)\)$. Actually the above defined sequence
$\delta=x_1<x_2<\cdots<x_s=R$ also depends on the number $\e$ although
we have not indicated this dependence.
 
Let us consider the sequence of numbers $\e_j=2^{-j}$, $j=1,2,\dots$,
We shall choose an appropriate sequence of numbers
$n_j=n_j(\e_j)\ge\bar n(\e_j)$, $j=1,2,\dots$ and construct the random
variables $\eta_k$ and $\bar\eta_k$ with the help of the same sequence
of numbers $\delta=x_1<x_2<\cdots<x_s=R$ considered in the previous
paragraph (which depends on $\e=\e_j$) for all indices $n_j\le k\le
n_{j+1}$. We shall show that in the case of a good choice of the
sequence $n_j$ and a good construction of the random variables
$\eta_k$ and $\bar\eta_k$, $k=1,2,\dots$ we can satisfy the statement
Proposition~2.

We shall construct the random variables $\eta_k$ and $\bar\eta_k$ with
infinitely divisible distributions by first constructing two
Poisson processes $\xi_{k,1},\xi_{k,2},\dots$ and
$\bar\xi_{k,2},\bar\xi_{k,2},\dots$ with counting measures $\mu_k$ and
$\mu_0$ respectively. Then we define the random variables $\eta_k$ and
$\bar\eta_k$ as the regularized sums of these Poisson processes
described in Part~I. More explicitly, we put
$$
\aligned
\eta_k(\oo)&=\lim_{N\to\infty}\(\sum_{p\colon\;|\xi_{k,p}(\oo)|\ge 2^{-N}}
\xi_{k,p}(\oo)-E\( \sum_{p\colon\;|\xi_{k,p}(\oo)|\ge 2^{-N}}
\tau(\xi_{k,p}(\oo)) \)\),\\
\bar\eta_k(\oo)&=\lim_{N\to\infty}
\(\sum_{p\colon\;|\bar\xi_{k,p}(\oo)|\ge 2^{-N}} \bar\xi_{k,p}(\oo)
-E\(\sum_{p\colon\;|\bar\xi_{k,p}(\oo)|\ge 2^{-N}}
\tau(\bar\xi_{k,p}(\oo))\)\),
\endaligned \tag2.4
$$
where the function $\tau(x)=\tau_a(x)$ was defined in formula~(1.2).
In Part~I we have seen that the limits in formula~(2,4) exist with
probability~1, and the random variables $\eta_k$ and $\bar\eta_k$ they
define  have the prescribed distributions.
 
To construct the Poisson processes $\xi_{k,1},\xi_{k,2},\dots$ and
$\bar\xi_{k,2},\bar\xi_{k,2},\dots$ with counting measures $\mu_k$
and $\mu_0$ respectively first we construct some  Poisson distributed
random variables $\zeta_{k,l}^{\pm}$ and $\bar\zeta_{k,l}^\pm$, $1\le
l<s$, from which the random variable $\zeta_{k,l}^+$ and
$\bar\zeta_{k,l}^+$. $l=1,2,\dots$,
tell us that the Poisson processes $\xi_{k,n}$ and $\bar\xi_{k,n}$,
$n=1,2,\dots$, how many points have in the interval $[x_l,x_{l+1})$.
Similarly, the random variables a $\zeta^-_{k,l}$  and
$\bar\zeta^-_{k,l}$ tell that these Poisson processes how many points
have in the intervals $[-x_{l+1},-x_{l})$.

To construct the above random variables and $\zeta_{k,l}^{\pm}$ and
$\bar\zeta_{k,l}^\pm$ let us first define independent Poisson random
variables $\alpha_{k,l}^\pm$, $\beta_{k,l}^\pm$, $1\le l<s$, with
Poisson distribution such that the distribution of
$\alpha^+_{k,l}$ has parameter $\min (\mu_k((x_l,x_{l+1})),
\mu((x_l,x_{l+1})))$ and the distribution of $\beta^+_{k,l}$ has
parameter
$$
\max(\mu_k((x_l,x_{l+1})),\mu_0((x_l,x_{l_1})))-\min
(\mu_k((x_l,x_{l+1}), \mu_0(x_l,x_{l+1}))).
$$
Similarly, let the distribution of $\alpha^-_{k,l}$ have parameter
$$
\min (\mu_k((-x_{l+1},-x_l)),\mu_0((-x_{l+1},-x_l))),
$$
and let the distribution of $\beta^-_{k,l}$ have parameter
$$
\align
&\max(\mu_k((-x_{l+1},-x_{l})),\mu_0((-x_{l+1},-x_l)))\\
&\qquad -\min (\mu_k((-x_{l+1},-x_{l})),\mu_0((-x_{l+1},-x_l))).
\endalign
$$
If $\mu_k((x_l,x_{l+1}))\le \mu_0((x_l,x_{l+1}))$, then put
 $\zeta_{k,l}^+=\alpha_{k,l}^+$,
$\bar\zeta_{k,l}^+=\alpha_{k,l}^++\beta_{k,l}^+$,
and if $\mu_k((x_l,x_{l+1}))> \mu_0((x_l,x_{l+1}))$, then put
$\bar\zeta_{k,l}^+=\alpha_{k,l}^+$,
$\zeta_{k,l}^+=\alpha_{k,l}^++\beta_{k,l}^+$, $1\le l<s$. Let us define
similarly the random variables $\zeta_{k,l}^-$ and $\bar\zeta_{k,l}^-$
only in this case we replace the interval
$(x_l,x_{l+1})$ by the interval $(-x_{l+1},-x_l)$ and the random
variables $\alpha^+_{k,l}$ and $\beta^+_{k,l}$
by the random variables $\alpha^-_{k,l}$ and $\beta^-_{k,l}$.
 
The random variables $\zeta_{k,l}^+$ and $\zeta_{k,l}^-$
are Poisson distributed, and their parameters are
$\mu_k((x_l,x_{l+1}))$ and $\mu_k((-x_{l+1},-x_l))$. Similarly, the
random variables $\bar\zeta_{k,l}^+$ and $\bar\zeta_{k,l}^-$ are
Poisson distributed with parameters $\mu_0((x_l,x_{l+1}))$ and
$\mu_0((-x_{l+1},-x_l))$. This means that the distributions of the
above constructed random variables agree with the distributions of
the number of points in the appropriate intervals of the Poisson
processes to be constructed. This property makes possible the
application of these random variables in the construction. Let us
also observe that these random variables also satisfy the identity
$$
\aligned
E\left|\zeta_{k,l}^+-\bar\zeta_{k,l}^+\right|=
\left|E\(\zeta_{k,l}^+-\bar\zeta_{k,l}^+\)\right|&
=\left|\mu_k((x_l,x_{l+1})) -\mu_k((x_l,x_{l+1}))\right|, \\
E\left|\zeta_{k,l}^--\bar\zeta_{k,l}^-\right|=
\left|E\(\zeta_{k,l}^--\bar\zeta_{k,l}^-\)\right|&=\left|
\mu_k((-x_{l+1},-x_l)) -\mu_k((-x_{l+1},-x_l))\right|.
\endaligned \tag2.5
$$
 
Now we turn to the construction of the Poisson processes $\xi_{k,n}$ and
$\bar\xi_{k,n}$, $n=1,2,\dots$, with the help of the above constructed
random variables $\zeta_{k,l}^\pm$ and $\bar\zeta_{k,l}^\pm$.
Let us throw $\zeta_{k,l}^+$ number of points to the interval
$(x_l,x_{l+1})$ and $\zeta_{k,l}^-$ number of points to the interval
$(-x_{l+1},x_l)$ independently of each other so that these points fall
into a set $A\subset (x_l,x_{l+1})$ or $A\subset (x_l,x_{l+1})$
with probability $\frac{\mu(A)}{\mu((x_l,x_{l+1}))}$ and
$\frac{\mu(A)}{\mu((x_{-l+1},-x_l))}$, $1\le l\le s$, respectively.
Similarly, let us consider the Poisson distributed random variable
$\zeta_{k,0}$ with parameter $\mu_k(-x_1,x_1)$ and the Poisson
distributed random variable $\zeta_{k,s}^++\zeta^-_{k,s}$ with
parameter $\mu_k((-\infty,-x_s)\cup(x_s,\infty))$, and let us throw
$\zeta_{k,0}$ number of points to the interval $(-x_1,x_1)$ so that a
point falls into a set $A\subset (-x_1,x_1)$  with probability
$\frac{\mu_k(A)}{\mu_k((-x_1,x_1))}$, and let us throw
$\zeta^+_{k,s}+\zeta^-_{k,s}$ number of points to the set
$(-\infty,-x_s)\cup(x_s,\infty)$ so that a point falls into a set
$A\subset (-\infty,-x_s)\cup(x_s,\infty)$ with probability
$\frac{\mu_k(A)}{\mu_k((-\infty,-x_s)\cup(x_s,\infty))}$. Let the
above considered random variables $\zeta_{k,l}$ together with all
point throws made in the above construction be independent of each
other. Then the union of the points thrown to different intervals is
a Poisson process $\xi_{k,1},\xi_{k,2},\dots$ with counting measure
$\mu_k$. We can construct similarly a Poisson process
$\bar\xi_{k,1},\bar\xi_{k,2},\dots$ with counting measure $\mu_0$
on the real line. Only in this case we replace the Poisson
distributed random variables $\zeta_{k,l}^\pm$, $0\le l\le s$, by
the Poisson distributed random variables $\bar\zeta_{k,l}^\pm$, $0\le
l\le s$, whose parameters can be given similarly, only the measure
$\mu_k$ is replaced by the measure $\mu_0$.
 
In such a way we have constructed the underlying Poisson processes and
the random variables $\eta_k$ and $\bar\eta_k$ determined by them.
(Only the threshold index $n_j=n_j(\e_j)$ is still not fixed.) We want
to show that the above construction satisfies the stochastic convergence
$\eta_k-\bar\eta_k\Rightarrow0$. The proof of this statement will be
based on the observations that the underlying Poisson processes
have almost the same number of points in the intervals $(x_{l-1},x_l)$,
(the relation (2.5) expresses such a fact). Beside this, these intervals
are very small, hence the precise position of the points falling to
them has a very small influence on the value of the random variables
$\eta_k$ and $\bar\eta_k$. To simplify further notations let us denote
by $B_l$ the interval $(x_l,x_{l+1})$ if $1\le l<s$ and the interval
$(x_{l-1},x_l)$ if $-1\ge l>-s$.
 
First we consider the contribution of those points of the Poisson
processes to the random variables $\eta_k$ and $\bar\eta_k$ which take a
large value, more explicitly whose absolute values are larger than $R$
with the number $R$ introduced in formula~(2.3). If $k\ge n_j\ge\bar
n(\e_j)$, then the measure $M_k$ satisfies relation~(2.3) with
$\e_j=2^{-j}$. This relation also holds if the measures $M_k$ are
replaced by the limit measure $M_0$. By the second relation of
formula~(2.3) the parameters of the Poisson distributed random
variables $\zeta_{k,s}=\zeta^+_{k,s}+\zeta^-_{k,s}$ and
$\bar\zeta_{k,s}=\bar\zeta^+_{k,s}+\bar\zeta^-_{k,s}$ are less than
$2^{-j}$. Hence the probability of the event that the corresponding
Poisson processes $\xi_{k,n}$ and $\bar\xi_{k,n}$, $n=1,2,\dots$,
contain no point such that $|\xi_{k,n}|>R$ or $|\bar\xi_{k,n}|>R$ is
greater than $1-2\cdot2^{-j}$. Furthermore, the expected number of the
points of the Poisson processes with absolute value larger than~$R$ is
less than $2\cdot2^{-j}$, and $|\tau(x)|\le a$ for all points $x\in
R^1$. The above relations imply that
$$  \allowdisplaybreaks
\align
&P\(\left|\sum_{\{n\colon\;|\xi_{k,n}|\ge
R\}}\xi_{k,n}-E\(\sum_{\{n\colon\;|\xi_{k,n}|\ge R\}}
\tau(\xi_{k,n})\)\right|\le 2a\cdot 2^{-j}\)<2\cdot
2^{-j} \quad\text{if }k\ge n_j, \\
&P\(\left|\sum_{\{n\colon\;|\bar\xi_{k,n}|\ge R\}}
\bar\xi_{k,n}-E\(\sum_{\{n\colon\;|\bar\xi_{k,n}|\ge R\}}
\tau(\bar\xi_{k,n})\)\right|\le2a\cdot 2^{-j}\)<2\cdot
2^{-j} \quad\text{if }k\ge n_j.
\tag2.6
\endalign
$$
 
For a Poisson process $\xi_n$, $n=1,2,\dots$, in the interval
$[a,b]$ with a finite counting measure $\mu$ the variance of the random
sum  $\summ_n \xi_n$ equals $\int_a^b u^2\mu(\,du)$, (see e.g.\ Lemma~1
in Part~I.) Hence the Chebishev inequality and the first part formula
(2.3) imply that for all sufficiently large $\delta>0$ (observe that
$\tau(x)=\tau_a(x)=x$ if $|x|\le \delta$.)
$$
\align
&P\(\left|\sum_{\{l\colon\;2^{-N}\le |\xi_{k,n}|\le\delta\}}
\xi_{k,n}-E\(\sum_{\{l\colon\;2^{-N}\le |\xi_{k,n}|\le\delta\}}
\tau(\xi_{k,n})\)\right|\ge 2^{-j}\)  \\
&\qquad \le 2^{2j} \text{Var}\,\(\sum_{\{l\colon\;2^{-N}\le
|\xi_{k,p}|\le\delta\}} \xi_{k,n}\)\le2^{2j}\e_j^3=2^{-j}\\
&P\(\left|\sum_{\{l\colon\;2^{-N} \le |\bar\xi_{k,n}|\le\delta\}}
\bar\xi_{k,n}-E\(\sum_{\{l\colon\;2^{-N} \le |\bar\xi_{k,n}|\le\delta\}}
\tau(\bar\xi_{k,n})\)\right|\ge2^{-j}\) \\
& \qquad \le 2^{2j} \text{Var}\,\(\sum_{\{n\colon\;2^{-N}\le
|\bar\xi_{k,n}|\le\delta\}}
\bar\xi_{k,n}\)\leq2^{2j}\e_j^3=2^{-j}\\
&\qquad\qquad\qquad\qquad\text{if }k\ge n_j \text{ and } 2^{-N}<\delta.
\tag2.7   \endalign
$$
Further we claim that if the indices $n_j$ are chosen sufficiently
large, then
$$ \allowdisplaybreaks
\align
P\Biggl(&\sum_{l\colon\; 1\le |l| <s}\(\sum_{n\colon\;\xi_{k,n}\in B_l}
\xi_{k,n}-E\(\sum_{n\colon\;\xi_{k,p}\in B_l} \tau(\xi_{k,n})\)\)
\tag2.8  \\
&\qquad-\(\sum_{n\colon\;\xi_{k,n}\in B_l}
\bar\xi_{k,n}-E\( \sum_{p\colon\;\xi_{k,n}\in
B_l}\tau(\bar\xi_{k,n})\)\)>2^{-j}\Biggr)
<2^{-j} \quad\text{if } k\ge n_j
\endalign
$$
 
First we show with the help of formulas (2.4), (2.6), (2.7) and (2.8)
that $\eta_k-\bar\eta_k\Rightarrow0$. Indeed, by summing up formulas
(2.6), (2.7) and (2.8) we get that of all integers $N$ such that
$2^{-N}<\delta$
$$ \allowdisplaybreaks
\align
P\Biggl(&\(\summ_{n\colon\; |\xi_{k,n}|\ge 2^{-N}}
\xi_{k,n}-E\(\summ_{n\colon\;|\xi_{k,n}|\ge 2^{-N}}\tau(\xi_{k,n})\)\)
\tag2.9   \\
&\quad-\(\summ_{n\colon\; |\bar\xi_{k,n}|\ge
2^{-N}}\bar\xi_{k,n}-E\(\summ_{n\colon\; |\xi_{k,n}|\ge 2^{-N}}
\tau(\bar\xi_{k,n})\)\)
>(4a+3)\cdot 2^{-j}\Biggr) <5\cdot 2^{-j}
\endalign
$$
if $k\ge n_j$.

Let us consider the $\liminf$ of the events whose probabilities were
investigated in formula (2.9) in the variable~$N$. (We recall that
$\liminff_{N\to\infty} A_N=\bigcupp_{N=1}^\infty
\(\bigcapp_{L=N}^\infty A_L\)$.) By applying formula (2.4) we get that
$P\(|\eta_k-\bar\eta_k|>(4a+3)\cdot2^{-j}\)<5\cdot
2^{-j}$ if $k\ge n_j$. Hence formula(2.8) implies the relation
$\eta_k-\bar\eta_k\Rightarrow0$ as we claimed.
To give the still missing proof of formula~(2.8) first we show that
$$
\aligned
P&\(\!\sum_{l\colon\; 1\le |l| <s}\!\(\!\sum_{n\colon\;\xi_{k,n}\in B_l}  \!\!
\xi_{k,n}-E\(\!\sum_{n\colon\;\xi_{k,n}\in B_l} \!\! \tau(\xi_{k,n})\)
-\zeta_{k,l}x_l+\tau(x_l)\mu_k(B_l)\)>2^{-2j}\) \\
&\qquad\qquad\qquad <2\cdot 2^{-2j} \quad\text{and} \\
P&\(\!\sum_{l\colon\; 1\le |l| <s}
\!\(\!\sum_{n\colon\;\bar\xi_{k,n}\in B_l}  \!\!
\bar\xi_{k,n}-E\(\!\sum_{n\colon\;\xi_{k,n}\in B_l}\!\!
\tau(\bar\xi_{k,n})\) -\bar\zeta_{k,l}x_l+\tau(x_l)\mu_0(B_l)\)>
2^{-2j}\) \\
&\qquad\qquad\qquad <2\cdot 2^{-2j}
\endaligned \tag2.10
$$
if $k\ge n_j$.
 
The first inequality of formula (2.10) bounds the error we commit if
those points of the Poisson processes $\xi_{k,n}$, $n=1,2,\dots$, for
which $\xi_{k,n}\in B_l$ are replaced by the end-point $x_l$ of the
interval $B_l$, and sum up these errors for all such points of the
Poisson process for which $\delta\le |\xi_{k,l}|\le R$. As
$$
\align
|\xi_{k,n}-x_l|&\le\supp_{1\le|l|\le s}|x_{l+1}-x_l|
\le\frac {\e_j^4}L,\\
\left|E\(\sum_{n\colon\;\xi_{k,n}\in
B_l}\tau(\xi_{k,n}^{(l)})\)-\tau(x_l)\mu_k(B_l)\right|
&\le\supp_{1\le|l|\le s}|x_{l+1}-x_l|\mu_k(B_l)\le
\frac {\e_j^4}L\mu_k(B_l)
\endalign
$$
if $k\ge n_j$, the first inequality of relation (2.10) follows from the
relations
$$
\align
&\sum_{l\colon\; 1\le |l| <s} \left|E\(\sum_{n\colon\; \xi_{k,n}\in
B_l}\tau(\xi_{k,n}^{(l)})\)-\tau(x_l)\mu_k(B_l)\right|\\
&\qquad\le \sum_{l\colon\; 1\le |l| <s}\frac {\e_j^4}L\mu_k(B_l)=
\e_j^4\frac{\mu_k((-R,R)\setminus(-\delta,\delta))}L\le \e_j^4
\endalign
$$
and
$$ \allowdisplaybreaks
\align
P&\(\sum_{l\colon\;1\le |l|<s}\left|\sum_{n\colon\; \xi_{k,n}\in B_l}
\xi_{k,n}-\zeta_{k,l}x_l\right|>\e_j^2 \)\\
&\qquad\le P\(\#\{n\colon\; \delta<|\xi_{k,n}|<R\}>\frac{L}{2\e_j^2}\)\le
\frac{2\e_j^2 E\(\#\{n\colon\; \delta<|\xi_{k,n}|<R\}\)}{L} \\
&\qquad =\frac{2\e_j^2\mu_k\((-R,-\delta)\cup(\delta,R)\)}{2L}\le
2\e_j^2=2\cdot2^{-2j}\quad \text{if }k\ge n_j.
\endalign
$$
The second inequality of formula (2.10) can be proved similarly, only
in this case we have to consider the Poisson process $\bar\xi_{k,n}$,
$n=1,2,\dots$, instead of the Poisson process $\xi_{k,n}$,
$n=1,2,\dots$, and the measure $\mu_k$ has to be replaced by the
measure $\mu_0$.
 
By formula (2.10) to complete the proof of formula (2.8) hence of
Proposition~2 it is enough to show that if the threshold indices
$n_j\ge \bar n_j$ are chosen sufficiently large, then
$$
P\(\sum_{l\colon\; 1\le |l|<s}
\left|x_l(\zeta_{k,l}-\bar \zeta_{k,l})-\tau(x_l)(\mu_k(B_l)-
\mu_0(B_l))\right|>2^ {-2j}\)\le
\frac{2^{-j}}2 \quad \text{if }k\ge n_j \tag2.11
$$
or to prove the following slightly stronger inequality:
$$
\sum_{l\colon\; 1\le |l|<s} \(|x_l|\,E|\zeta_{k,l}-\bar \zeta_{k,l}|+
|\tau(x_l)|\,|\mu_k(B_l)-\mu_0(B_l)|\)<2^ {-4j}\quad\text{if }
k\ge n_j. \tag2.12
$$
 
But formula (2.5) implies that
$E|\zeta_{k,l}-\bar\zeta_{k,l}|=C_j|\mu_k(B_l)-\mu_0(B_l)|$.
Beside this, the identities $\limm_{k\to\infty}\mu_k(B_l)=\mu_0(B_l)$
hold because of the (weak) convergence of the canonical measures $M_k$
and $M_0$. As the sum in formula (2.12) contains only finitely many
terms (the number of terms depends only on the index $j$), and each
term tends to zero as $k\to\infty$ by the above observations the
expression at the left-hand side of formula (2.12) can be made an
arbitrary small positive number by choosing the threshold index $n_j$
sufficiently large.
 
\beginsection 3. The functional limit theorem. The proof of Theorem 2.
 
First we shall show with the help of Lemma~1 and Proposition~1 that
also the stochastic processes $S_k(t)$, $0\le t\le1$, appearing in
Theorem~2 can be split to two parts, one of them responsible for the
convergence of the Gaussian and one of them responsible for the
convergence of the Poissonian part. The convergence of the Gaussian
part can be deduced from the functional central limit theorem and the
convergence of the Poissonian part can be reduced to a simpler
statement about the convergence of infinitely divisible processes. This
will be the content of Part~A in Section~3. In Part~B we prove the
convergence of the Poissonian part with the help of two Propositions.
The first of them, Proposition~3, enables us to discretize the time
parameter of the stochastic processes we investigate. Proposition~4
gives a good coupling of infinitely divisible processes whose
canonical measures (on the strip $\bold R^1\times[0,1]$) are close to
each other. It can be considered as a generalization of Theorem~2 where
the coupling of infinitely divisible processes is considered instead
of the coupling of infinitely divisible random variables. Finally in
Part~C we prove Propositions~3 and~4, and this completes the proof
of Theorem~2.
 
\medskip\noindent
{\script A.) Poisson approximation. Separation of the normal and
Poisson part of the limit process.} \medskip
 
We reduce the proof of Theorem~2 to that of a simpler statement with
the help of Lemma~1, Proposition~1 and the coupling construction
described after Lemma~1.
 
Let us consider the random variables $\xi'_{k,j}$, $\xi''_{k,j}$,
$\eta'_{k,j}$, $\eta''_{k,j}$, $\tilde\xi_{k,j}=\xi_{k,j}'+\xi''_{k,j}$
and $\eta_{k,j}=\eta_{k,j}'+\eta''_{k,j}$, $1\le j\le n_k$, introduced
in the coupling construction described after Lemma~1 together with the
partial sums $S'_{k,l}=\summ_{j=1}^l\xi'_{k,j}$, $S''_{k,l}=
\summ_{j=1}^l\xi''_{k,j}$, $\tilde S_{k,l}=S'_{k,l}+S''_{k,l}$,
$T'_{k,l}=\summ_{j=1}^l(\eta'_{k,j}-E\eta'_{k,j})$,
$T''_{k,l}=\summ_{j=1}^l(\eta''_{k,j}-E\tau(\eta''_{k,j}))$,
$T_{k,l}=T'_{k,l}+T''_{k,l}$, $1\le l\le n_k$. Let us also introduce the
stochastic processes $S'_k(t)$, $S''_k(t)$, $\tilde S_k(t)$, $T'_k(t)$,
$T''_k(t)$ and $T_k(t)$, $0\le t\le 1$, with cadlag function
trajectories which we define similarly to the stochastic process
$S(t)$, $0\le t\le1$, introduced in formula (1.12) with the difference
that we replace the random variables $S_{k,l}$ in formula (1.12) by
the random variables $S'_{k,l}$, $S''_{k,l}$, $\tilde S_{k,l}$ and
$T'_{k,l}$, $T''_{k,l}$, $T_{k,l}$. It follows from formula (1.8)
and the identity $E\xi'_{k,j}+E\tau(\xi''_{k,j})
=E\tau(\tilde\xi_{k,j})=0$ that
$$
\supp_{0\le t\le 1}|T_k(t)-\tilde S_k(t)|\Rightarrow0,\quad \text {if }
k\to\infty, \tag3.1
$$
where $\Rightarrow$ denotes stochastic convergence. The distributions
of the stochastic processes $S_k(t)$ and $\tilde S_k(t)$ agree.
Furthermore, if $x_k(\cdot)$ and $y_k(\cdot)$ are such functions in the
space $D([0,1])$ for which $\limm_{k\to\infty}\supp_{0\le t\le 1}
|x_k(t)-y_k(t)|=0$ then also the relation
$\limm_{k\to\infty}d(x_k(\cdot),y_k(\cdot))=0$ holds with the metric
$d(\cdot,\cdot)$ introduced at page~10 to metrize the space $D([0,1])$.
Thus by Theorem~A and formula~(3.1) to prove Theorem~2 it is enough to
show that the stochastic processes $T_k(t)$, $0\le t\le1$,
converge weakly to the limit stochastic process described in Theorem~2.
 
The identity $T_k(t)=T'_k(t)+T''_k(t)$ holds, and the stochastic
processes $T'_k(t)$ and $T''_k(t)$, $1\le t\le 1$, appearing in this
formula are independent. We shall show that the stochastic processes
$T_k'(\cdot)$ converge to the Gaussian and the stochastic processes
$T''_k(\cdot)$ converge to the Poisson component of the limit process
as $k\to\infty$. To prove these statements first we have to clarify how
the convergence of the canonical measures $N_k$ on the strip $\bold
R^1\times[0,1]$ to a canonical measure $N_0$ is reflected in the
behaviour of the canonical measures corresponding to the stochastic
processes $T'_k(\cdot)$ and $T''_k(\cdot)$.
 
Let us consider a sequence of positive numbers $\e_k$ with which
Lemma~1 holds. Let us define, by using the notation of Lemma~1,
the measures $N'_k$ on the interval $[0,1]$ concentrated in
the points $0\le u_{k,1}\le u_{k,2}\le\cdots\le u_{k,n_k}=1$ 
introduced in the formulation of Theorem~2, for which $N'_k(u_{k,l})
=G_{k,l}(\e_k)-G_{k,l}(-\e_k)$, $k=1,2,\dots$, where
$G_{k,l}(\,dx)=x^2F_{k,l}(\,dx)$, similarly to the formulation of
Lemma~1. Let us also define the
canonical measures $N''_k$ on the strip $\bold R^1\times[0,1]$ which are
concentrated on the union of the lines $\bold R^1\times u_{k,l}$, $1\le
l\le n_k$, and
$$
N''_k(B\times \{u_{k,l}\})=\int_{B\cap \{u\colon\;|u|\ge\e_k\}}
u^2F_{k,l}(\,du), \quad 1\le l\le n_k, \quad k=1,2,\dots.
$$
We claim that under the conditions of Theorem~2 the measures $N_k'$
weakly converge to the measure $N_0'$ defined by the relation
$N_0'(B)=N_0(\{0\}\times B)$ if $B\subset [0,1]$. Beside this, the
canonical measures $N''_k$ on the strip $\bold R^1\times [0,1]$
converge weakly to the canonical measure $N''_0$ defined by the
relation $N''_0(B)=N_0(B\setminus(\{0\}\times [0,1]))$ if $B\subset
\bold R^1\times[0,1]$.
 
To prove the above statements let us observe that if $B\subset [0,1]$
is a set with boundary $\partial B$ such that $\lambda(\partial B)=0$
and $C\subset\bold R^1$ is a bounded set such that $M_0(\partial B)=0$,
then under the conditions of Theorem~2 $\limm_{k\to\infty}N_k(C\times
B)=N_0(C\times B)$. Furthermore, $\limm_{k\to\infty}N'_k([0,1])
=M(\{0\})$  by Lemma~1. First we show that for an arbitrary set
$B\subset [0,1]$ whose boundary satisfies the relation
$\lambda(\partial B)=0$, $\limsupp_{k\to\infty}N'_k(B)\le
N_0(\{0\}\times B)=N'_0(B)$. Indeed, for all numbers $\delta>0$ there
exists an interval $[-\eta,\eta]$ with some $\eta>0$ such that
$\pm\eta$ is a point of continuity of the measure $M$ and
$M_0([-\eta,\eta])\le M_0(\{0\})+\delta$, and this implies that
$N_0([-\eta,\eta]\times B)\le N_0(\{0\}\times B)+\delta$. Since
$\e_k\to0$ if $k\to\infty$, it follows from the above facts that
$\limsupp_{k\to\infty}N'_k(B)\le \limm_{k\to\infty}N_k(B\times
[-\eta,\eta])=N_0([-\eta,\eta]\times B)\le N_0(\{0\}\times B)+\delta$.
As this relation holds for all numbers $\delta>0$, hence
$\limsupp_{k\to\infty} N_k'(B)\le N'_0(B)$. By applying this inequality
for both sets $B$ and $[0,1]\setminus B$ we get that
$$
\align
N'_0([0,1])=&M_0(\{0\})=
\limm_{k\to\infty}N'_k([0,1])\le\limsupp_{k\to\infty}
N_k'(B)+\limsupp_{k\to\infty}N_k'([0,1]\setminus B)\\
&\le N'_0(B)+N'_0([0,1]\setminus B))=N_0'([0,1]).
\endalign
$$
This series of inequalities may be valid only if also the identity
$\limm_{k\to\infty} N'_k(B)=N'_0(B)$ holds.
 
Let us also introduce the canonical measures $\bar N_k'$, $k=1,2,\dots$
and $\bar N_0'$ on the strip $\bold R^1\times [0,1]$ with the help of
the formulas $\bar N'_k(A)=N_k(A\cap[-\e_k,\e_k])$, $\bar
N_0'(A)=N_0(A\cap \{0\}\times[0,1])$, $A\subset \bold R^1\cap [0,1]$.
(In these formulas we lifted the measures $N'_k$, $k=1,2,\dots$, to the
strip $\bold R^1\times [0,1]$.) Then the convergence of the measures
$N_k'$ to the measure $N_0'$ implies the convergence of the canonical
measures $\bar N_k'$ to the canonical measure $\bar N_0'$. Furthermore,
$N''_k=N_k-\bar N'_k$, $k=1,2,\dots$, and the convergence of the
canonical measures $N''_k$ on the strip $\bold R^1\times [0,1]$
follows from the facts that the canonical measures $N_k$ converge to
the canonical measure $N_0$ and the canonical measures converge to the
canonical measure $\bar N_k'$.
 
Let us define the random variables $T'_{k,l}
=\summ_{j=1}^{l}\eta'_{k,j}$, $1\le l\le n_k$, $T'_{k,0}=0$ and  the
stochastic processes $T'_k(t)$, $T'_k(t)=T'_{k,l}$ if $u_{k,l-1}\le
t<u_{k,l}$, $T_{k,l}(1)=T'_{k,n_k}$. Let us also consider the
continuous function $\lambda(t)=N_0(\{0\}\times [0,t])$, $0\le t\le 1$
introduced in Part~a) of Theorem~2. We claim that the stochastic
processes $T'_k(t)$, $0\le t\le1$, weakly converge to the stochastic
process $W(\lambda(t))$, $0\le t\le1$, as $k\to\infty$, where $W(t)$,
$0\le t\le M(\{0\})$, is a standard Wiener process.
 
To prove the above statement let us introduce the numbers
$\bar u_{k,0}=0$, $\bar u_{k,l}=\frac 1{U_k}\summ_{j=1}^l \text
{Var}\, \eta_{k,j}'$, $1\le l\le n_k$, $k=1,2,\dots$, where
$U_k=\summ_{j=1}^{n_k} \text{Var}\, \eta'_{k,j}$, together with the
stochastic processes $\bar T_{k}'(\cdot)$ which will be defined
similarly to the stochastic processes $T'_{k}(\cdot)$ with the only
difference that the numbers $u_{k,l}$ are replaced by the numbers $\bar
u_{k,l}$ in the definition. Then we can state on the basis of the
functional central limit theorem that the stochastic processes $\bar
T'_{k}(t)$, $0\le t\le1$, converge weakly to a standard Wiener
process~$W(t)$ as $k\to\infty$.
 
 
Let us define the monotone increasing and continuous functions
$\lambda_k(t)$, $k=1,2,\dots$, so that $\lambda_k(u_{k,l})=\bar
u_{k,l}$, $0\le l\le k_n$, and the function $\lambda_k(\cdot)$ is
linear in the intervals $[u_{k,l-1}, u_{k,l}]$, $1\le k\le n_k$. We
shall show that
$$
\limm_{k\to\infty}U_k=M_0(\{0\}), \quad
\limm_{k\to\infty}\supp_{0\le t\le1}\left|\lambda_k(t)
-\frac{\lambda(t)}{M_0(\{0\})}\right|=0, \tag 3.2
$$
and this implies that the stochastic processes $T_k'(t)$, $0\le t\le1$
weakly converge to the stochastic process $\sqrt {M_0(\{0\})} W\(\frac
{\lambda(t)}{M_0(\{0\})}\)$ whose distribution agrees with the
distribution of the stochastic process $W(\lambda(t))$, $0\le t\le 1$.
 
Indeed, Lemma~1 and formula (2.2) imply that
$$
\limm_{k\to\infty}U_k=\limm_{k\to\infty}\summ_{j=1}^{n_k}
E{\xi'_{k,j}}^2=M(\{0\}).
$$
We get similarly that $\limm_{k\to\infty}\supp_{1\le
l\le n_k}\left|\bar u_{k,l}U_k-\summ_{j=1}^lE{\xi'_{k,j}}^2\right|=0$,
that is
$$
\limm_{k\to\infty}\supp_{1\le l\le n_k}\left|\bar
u_{k,l}M_0(\{0\})-N_k([-\e_k,\e_k])\times[0,u_{k,l}])\right|=0.
$$
The monotone functions $h_k(t)=N_k([-\e_k,\e_k])\times[0,t])$ converge
to the monotone, continuous function $\lambda(t)=N_0(\{0\}\times
[0,t])$ for all numbers $0\le t\le1$, and because the above functions
are monotone, and the limit function is continuous the above
convergence is uniform. Hence $\limm_{k\to\infty}\supp_{1\le l\le
n_k}\left|\bar u_{k,l}M_0(\{0\})-\lambda(u_{k,l})\right|=0$. Thus
$$
\limm_{k\to\infty}\supp_{1\le l\le n_k}\left|\lambda_k(
u_{k,l})-\frac{\lambda(u_{k,l})}{M_0(\{0\})}\right|=0.
$$
This implies relation (3.2).

To prove the weak convergence of the stochastic processes
$T'_k(t)=\frac1{\sqrt{U_k}}\bar T'_k\(\lambda_k(t)\)$
to the stochastic process $W(\lambda(t))$ it is enough to show that
$$
\sqrt{U_k}\bar T'_k\(\lambda_k(t)\)-
\sqrt{U_k}\bar T'_k\(\frac{\lambda(t)}{M_0(0)}\)\Rightarrow0, \tag3.3
$$
where $\Rightarrow$ denotes stochastic convergence. Indeed, Theorem~A
and relation (3.3) imply the desired statement, since the stochastic
processes $\sqrt{U_k}\bar T'_k\(\frac{\lambda(t)}{M_0(0)}\)$ converge
weakly to the stochastic process $W(\lambda(t))$. On the other hand,
relation (3.3) follows from relation (3.2) and the result of the
general theory by which the weak convergence of the stochastic
processes $\sqrt{U_k}\bar T'_k\(\frac{\lambda(t)}{M_0(0)}\)$ to a
stochastic process with continuous trajectories follows that the
distributions of these processes are uniformly tight, i.e.\ for all
numbers $\e>0$ and $\eta>0$ there exists a number
$\delta=\delta(\e,\eta)>0$ such that
$$
P\(\sup_{(s,t)\colon\;|t-s|\le\delta} \left|\sqrt{U_k}\bar
T'_k\(\frac{\lambda(t)}{M_0(0)}\)-
\sqrt{U_k}\bar T'_k\(\frac{\lambda(s)}{M_0(0)}\)\right|>\eta\)\le\e
$$
for all indices $k=1,2,\dots$. (The number $\delta=\delta(\e,\eta)$
does not depend on the index~$k$.)
 
 
\medskip\noindent
{\script B.) The method of the proof. The study of the convergence of
the Poissonian part.} \medskip
 
In Part~A of Section~3 we defined the representations
$T_k(t)=T'_k(t)+T''_k(t)$ of the stochastic processes $T_k(\cdot)$,
$1\le t\le 1$, $k=1,2,\dots$, introduced there and showed that Theorem~2
follows from the convergence of the distributions of the stochastic
processes $T_k(t)$, $k=1,2,\dots$, to the process $S(t)$, defined in the
formulation of Theorem~2, in the space $D([0,1])$. Beside this, the
stochastic processes $T'_k(t)$ and $T''_k(t)$ are independent, and
the stochastic processes $T'_k(t)$ converge weakly to a Gaussian
process with independent increments whose distribution can be described
similarly to the limit process given in Theorem~2 with the difference
that the measure $N_0(\cdot,\cdot)$ on the strip $\bold
R^1\times[0,1])$ is replaced by the measure $N_0'(A)=N_0(A\cap \{0\})$
in formula~(1.15), or more explicitly in the definition of the measure
$M_{0,u,v}$ appearing in this formula. Hence to complete the proof of
Theorem~2 it is enough to show that the stochastic processes
$T''_k(t)$ converge weakly to a stochastic process $S''_0(t)$, $0\le
t\le1$, with independent increments described by the canonical measure
$N_0''(\cdot,\cdot)$ on the strip $\bold R^1\times[0,1]$, given by the
formula $N_0''(A)=N_0(A\cap (\bold R^1\setminus\{0\})\times[0,1])$. To
simplify further discussions let us introduce the following definition.

\medskip\noindent
{\bf The definition of the (time)--discretization of a stochastic
process.} {\it Let $Z(t)$, $0\le t\le1$, be a stochastic process on the
interval\/ $[0,1]$, and let $0=t_0<t_1<t_2<\cdots<t_s=1$, be a monotone 
sequence in the interval\/ $[0,1]$. Then the
discretization of the stochastic process $Z(t)$, $0\le t\le1$,
determined by the sequence of numbers $0=t_0< t_1<t_2<\cdots<t_s=1$ is
the stochastic process $\bar Z(t)=\bar Z_{t_0,t_1,\dots,t_s}(t)$, $0\le
t\le1$, given by the formula
$$
\bar Z(t)=\bar Z_{t_0,t_1,\dots,t_s}(t)=Z(t_{l-1}), \quad\text{if }
t_{l-1}\le t<t_l, \; 1\le l\le s,\quad \bar Z(1)=Z(1).
$$
\medskip}
 
Let us observe that the stochastic processes $T''_k(t)$ are
discretizations (in the points $0=u_{0,k}\le u_{1,k}\le\cdots\le
u_{k,n_k}=1$,) of infinitely divisible stochastic processes determined
by Poisson processes with such counting measures
$\nu_k(dx,dy)=\frac{N_k(dx,dy)}{x^2}$ for which the canonical measures
$N_k$ on the strip $\bold R^1\times[0,1]$ converge weakly to a
canonical measure $N''_0$, and the identity $N_k(\{0\}\cap[0,1])=0$
holds. Hence we complete the proof of Theorem~2 if we prove the {\it
Statement}\/ formulated below. It can be considered as a
generalization of Proposition~2 to random variables taking values in
more general function spaces.

\medskip\noindent
{\bf Statement.} {\it Let $N_k$, $k=0,1,2,\dots$, be a sequence of
canonical measures on the strip $\bold R^1\times[0,1]$, and assume that
these canonical measures $N_k$ converge weakly to a canonical measure
$N_0$ if $k\to\infty$, they satisfy condition~b.) of Theorem~2, and
$N_k(\{0\}\times [0,1])=0$, $k=0,1,2,\dots$. (This latter condition
means that neither the processes determined by the measures $N_k$,
$k=1,2,\dots$, nor the process determined by the limit measure $N_0$
have a Gaussian component.) Let us define the canonical measures
$\nu_k(dx,dy)=\frac{N_k(dx,dy)}{y^2}$, $k=0,1,2,\dots$, and
consider Poisson fields $X_n(k)=(X_n^{(1)}(k),X_n^{(2)}(k))$,
$X_n(k)\in \bold R^1\times[0,1]$, $n=1,2,\dots$, $k=1,2,\dots$,
on the strip $\bold R^1\times [0,1]$ with canonical measures
$\nu_k(dx,dy)$. Let us consider the infinitely divisible stochastic
processes $T_k(t)$, $0\le t\le1$, $k=0,1,2,\dots$, determined by these
stochastic fields which can be considered as $D([0,1])$ space valued
random variables. The distributions of the stochastic processes
$T_k(t)$, $k=1,2,\dots$, converge to the distribution of the stochastic
process $T_0(t)$ in the space $D([0,1])$ as $k\to\infty$.
 
Beside this, let us have for all numbers $k=1,2,\dots$ a partition
$0=u_{k,0}<u_{k,1}<\cdots<u_{k,n_k}=1$ of the interval $[0,1]$, which
satisfy the condition $\supp_{1\le j\le n_k}|u_{k,j}-u_{k,j-1}|=0$, and
let us consider the discretizations $\bar T_k(t)=\bar
T_{k,u_{k,0},u_{k,1},\dots,u_{k,n_k}}(t)$ of the infinitely divisible
processes $T_k(t)$. The distributions of these discretizations $\bar
T_k(t)$ of the stochastic processes $T_k(t)$ also converge weakly to
the distribution of the stochastic process $T_0(t)$ in the space
$D([0,1])$.}\medskip
 
The missing part of Theorem~2 agrees with the second part of the {\it
Statement}\/ about the convergence of the stochastic processes
$\bar T_{k,u_{k,0},u_{k,1},\dots,u_{k,n_k}}(\cdot)$ to the stochastic
process $T_0(\cdot)$ if the same sequences of numbers
$0=u_{k,0}<u_{k,1}<\cdots<u_{k,n_k}=1$ are considered as in Theorem~2.
 
Now we formulate two propositions make some comments about them and give
the proof of the {\it Statement}\/ with their help. These propositions
will be proved in Part~C.. Before their formulation let us recall how
an infinitely divisible stochastic process (with nice trajectories in
the space $D([0,1])$ can be constructed by means of a Poissonian field
with a counting measure which has some nice properties.

Let $X_n=(X_n^{(1)},X_n^{(2)})$, $X_n\in \bold R^1\times[0,1]$,
$n=1,2,\dots$, be a Poisson field with such a counting measure $\nu$
on the strip $\bold R^1\times[0,1]$ for which $\nu(\bold R^1\setminus
[-b,b]\times[0,1])<\infty$ and $\int_{(x,y)\colon\;|x|\le b}x^2
\nu(\,dx,dy)<\infty$ for all numbers $b>0$. Then this Poisson field
determines an infinitely divisible stochastic process $T(t)$, $0\le
t\le1$, defined in the following way: Let us choose an appropriate
sequence of numbers $A_L$, $L=1,2,\dots$, $\limm_{L\to\infty}A_L=0$,
and put
$$
T^{(L)}(t)=\!\!\summ_{n\colon\;|X_n^{(1)}|>A_L,\,0\le X_n^{(2)}\le
t}X_n^{(1)}\!\!- E \(\!\summ_{n\colon\;|X_n^{(1)}|>A_L,\,0\le X_n^{(2)}\le
t}\! \tau\( X_n^{(1)}\)\), \quad 0\le t\le1,
$$
$L=1,2,\dots$, where the function $\tau(\cdot)$ was defined in formula
(1.2). It is proven in Lemma~2 in Section~5 of Part~I that if the
sequence $A_L$ tends to zero sufficiently fast, then the limit
$T(t)=\limm_{L\to\infty}T^{(L)}(t)$, $0\le t\le 1$, exists with
probability 1, where the limit is taken in supremum norm in the
interval $[0,1]$. Beside this, the trajectories of the so constructed
stochastic process $T(t)$ is an infinitely divisible stochastic process
with cadlag trajectories. If $N_0$ is a canonical measure on the strip
$\bold R^1\times[0,1]$, then the distribution of the increments of the
above defined stochastic process $T(t)$, $0\le t\le1$, determined by a
Poisson field with counting measure $\nu_0(dx,dy)
=\frac{N_0(dx,dy)}{x^2}$ is described by formula (1.15). We shall call
the stochastic process $T(t)$, $0\le t\le1$, constructed in the above
way the infinitely divisible process determined by the Poisson field
$X_n=(X_n^{(1)},X_n^{(2)})$, $X_n\in \bold R^1\times[0,1]$,
$n=1,2,\dots$. 

\medskip\noindent
{\bf  Proposition~3.} {\it Let a canonical measure $N_0$ be given on
the strip $\bold R^1\times [0,1]$ such that $N_0(\{0\}\times [0,1])=0$.
Let us also assume that the measure $N_0$ satisfies Condition~b.) of
Theorem~2. Let us consider a Poisson field $X_n=(X_n^{(1)},X_n^{(2)})$,
$X_n\in \bold R^1\times[0,1]$, $n=1,2,\dots$, on the strip $R^1\times
[0,1])$ with counting measure $\nu_0(dx,dy)=\frac{N_0(dx,dy)}{x^2}$,
and let $T_0(t)$, $0\le t\le1$, be the infinitely divisible process
determined by this Poisson field. For all numbers $\e>0$ and $\eta>0$
there exists a number $=\delta(\e,\eta)$ such that for all sequences of
number $0=t_0<t_1<\cdots<t_s=1$ for which the inequality $\supp_{1\le
l\le s}|t_l-t_{l-1}|<\delta$ holds the stochastic process $T_0(t)$ and
its discretization $\bar T_0(t)=\bar T_{0,t_0,t_1,\dots,t_s}(t)$
satisfies the inequality
$$
P\(d(T_0(\cdot), \bar T_0(\cdot))>\eta\)<\e, \tag3.4
$$
where $d(\cdot,\cdot)$  denotes  the (simpler, not complete) metric
introduced in the space $D([0,1])$.
 
Let $N_k$, $k=1,2,\dots$, be canonical measures on the strip $\bold
R^1\times [0,1]$ which satisfy the relation $N_k(\{0\}\times [0,1])=0$
and which converge to the canonical measure $N_0$ considered in the
previous paragraph. Let us define the measures
$\nu_k(dx,dy)=\frac{N_k(dx,dy)}{x^2}$, $k=1,2,\dots$ and consider
Poisson fields $X_n(k)=(X_n^{(1)}(k),X_n^{(2)}(k))$, $X_n(k)\in \bold
R^1\times[0,1]$, $n=1,2,\dots$, $k=1,2,\dots$, on the strip $R^1\times
[0,1])$ with counting measures $\nu_k(dx,dy)$. Let $T_k(t)$, $0\le
t\le1$, $k=1,2,\dots$, denote the infinitely divisible stochastic
processes determined by these Poissonian fields. Given some numbers
$\e>0$ and $\eta>0$ there exists a number $\delta=\delta(\e,\eta)$ and
a threshold index $k_0=k_0(\eta,\e)$ such that for all indices $k\ge
k_0$ and sequences of numbers $0=t_0<t_1<\cdots<t_s=1$ for which the
inequality $\supp_{1\le l\le s}|t_l-t_{l-1}|<\delta$ holds the
stochastic processes $T_k(t)$ and their discretizations $\bar
T_k(t)=\bar T_{k,t_0,t_1,\dots,t_s}(t)$ satisfy the inequality
$$
P\(d(T_k(\cdot), \bar T_k(\cdot))>\eta\)<\e\quad\text{if }k\ge k_0,
\tag3.5
$$
where $d(\cdot,\cdot)$ is the same metric in the space $D([0,1])$ as
that in formula (3.4).} \medskip
 
Let us emphasize that the threshold index $k_0=k_0(\eta,\e)$ in formula
(3.5) can be chosen independently of the (sufficiently dense) sequence
of numbers $0=t_0<t_1<\cdots<t_s=1$.

\medskip\noindent
{\bf Proposition~4.} {\it Let a sequence of canonical measures $N_k$,
$k=1,2,\dots$, be given on the strip $\bold R^1\times [0,1]$ which
converges to a canonical measure $N_0$ as $k\to\infty$, and
$N_k(\{0\}\times[0,1])=0$ for all numbers $k=0,1,2,\dots$. Furthermore,
let us fix a finite, monotone increasing sequence 
$0=t_0<t_1<t_2<\cdots<t_s=1$ on the interval
$[0,1]$. Then for all indices $k=1,2,\dots$ a Poisson field
$X_n(k)=(X_n^{(1)}(k),X_n^{(2)}(k))$, $X_n(k)\in \bold R^1\times[0,1]$,
$n=1,2,\dots$, can be constructed with canonical measures
$\nu_k(\,dx,dy)=\frac{N_k(\,dx,\,dy)}{x^2}$ together with a Poisson
field $X'_n(k)=({X'_n}^{(1)}(k), {X'_n}^{(2)}(k))$, $X'_n(k)\in \bold
R^1\times[0,1]$, $n=1,2,\dots$, with canonical measures
$\nu_0(\,dx,\,dy)=\frac{N_0(\,dx,\,dy)}{x^2}$ in such a way that the
infinitely divisible stochastic processes $T_k(t)$ and $T'_k(t)$,
$k=1,2,\dots$, determined by these Poisson fields, or more explicitly
their discretizations, the stochastic processes $\bar T_k(t)=\bar
T_{k,t_0,t_1,\dots,t_s}(t)$ and $\bar T'_k(t)=\bar
T'_{k,t_0,t_1,\dots,t_s}(t)$, $0\le t\le1$, satisfy the relation
$$
\sup_{0\le t\le 1}|\bar T_k(t)-\bar T_k'(t)|\Rightarrow0 \quad
\text{if } k\to\infty,  \tag3.6
$$
where $\Rightarrow$ denotes stochastic convergence. (Let us remark
that the distributions of the stochastic processes $T'_k(\cdot)$ and of
their discretizations do not depend on the index~$k$, since they are
determined by a Poisson field with canonical measure~$\nu_0$.)}
\medskip
 
Let a sequence of canonical measures $N_k$, $k=1,2,\dots$, be given on
the strip $\bold R^1\times[0,1]$ which satisfies the conditions of the
{\it Statement.}\/ Then with the help of Propositions~3 and~4 for all
numbers $\e>0$ and $\eta>0$ a partition $0=t_0<t_1<t_2<\cdots<t_s=1$ of
the interval $[0,1]$ can be given together with two sequences of
Poisson fields $X_n(k)=({X_n}^{(1)}(k),{X_n}^{(2)}(k))$ and
$X'_n(k)=({X'_n}^{(1)}(k),{X'_n}^{(2)}(k))$, $n=1,2,\dots$,
$k=1,2,\dots$ on the strip $\bold R^1\times [0,1])$ with counting
measures $\nu_k(dx,dy)= \frac{N_k(dx,dy)}{x^2}$ and
$\nu_0(dx,dy)=\frac{N_0(dx,dy)}{x^2}$ respectively which satisfy the
following property. The infinitely divisible stochastic processes
$T_k(t)$ and $T'_k(t)$, $0\le t\le1$, determined by the Poisson fields
$X_n(k)$ and $X'_n(k)$, $n=1,2,\dots$, $k=1,2,\dots$, and their
discretizations, the stochastic processes $\bar T_k(\cdot)=\bar
T_{k,t_0,t_1,\dots,t_s}(\cdot)$ and $\bar T'_k(\cdot)=\bar
T'_{k,t_0,t_1,\dots,t_s}(\cdot)$, satisfy the relations
$$
\aligned
P\(d(T_k'(\cdot),\,\bar T'_{k,t_0,\dots,t_s}(\cdot))>\eta\)&<\e \quad
\text{for all numbers } k\ge1 \\
P\(d(T_k(\cdot),\,\bar T_{k,t_0,\dots,t_s}(\cdot))>\eta\)&<\e,\quad
\text{if } k\ge k_0\\
P\(\sup_{0\le t\le 1}\left|\bar T_{k,t_0,\dots,t_s}(t)
-\bar T'_{k,t_0,\dots,t_s}(t)  \right|>\eta\)&<\e, \quad\text{if }
k\ge k_0,
\endaligned \tag3.7
$$
where $k_0=k_0(\e,\eta)$ is an appropriate threshold index.
 
Indeed, by Proposition~3 the first two relations of formula (3.7) hold
for an appropriate partition $0=t_0<t_1<t_2<\cdots<t_s=1$ of the
interval for all indices $k\ge k_0$ if $k_0=\bar k_0(\e,\eta)$ is
sufficiently large. The validity of these relations does not depend on
the way the Poisson fields $X_n(k)$ and $X'_n(k)$, $n=1,2,\dots$,
$k=1,2,\dots$, with counting measures $\nu_k$ and $\nu_0$ are
constructed. Then we can guarantee, because of Proposition~4, with
an appropriate construction that also the third relation of
formula~(3.7) holds. (In this step we may increase the threshold
index~$k_0$ if it is needed.)
 
Let us apply formula (3.7) with numbers $\e_j=\eta_j=\frac1j$.
Then we can see that there exists a monotone sequence of positive
integers $k_0\(\frac1j\)$ and for all $j=1,2,\dots$ a sequence
of numbers $0=t_0^{(j)}<t_1^{(j)}<t_2^{(j)}<\cdots<t_{s_j}^{(j)}=1$
can be given which satisfy formula (3.7) for all $j=1,2,\dots$ with
the choice $\e=\eta=\frac1j$ where the condition $k\ge k_0$
is replaced by the condition $k_0\(\frac1j\)\le k\le
k_0\(\frac1{j+1}\)$, and we write
$0=t_0^{(j)}<t_1^{(j)}<t_2^{(j)}<\cdots<t_{s_j}^{(j)}=1$ instead of
$0=t_0<t_1<t_2<\cdots<t_s=1$, i.e.\ the partition of the interval
$[0,1]$ we consider may depend on the index~$j$. Hence under the
conditions of the {\it Statement}\/ two sequences of Poisson fields
$X_n(k)$ and $X'_n(k)$, $n=1,2,\dots$, $k=1,2,\dots$, can be constructed
on the strip $\bold R^1\times[0,1]$ with canonical measures $\nu_k$ and
$\nu_0$ in such a way that the infinitely divisible stochastic
processes $T_k(t)$ and $T'_k(t)$, $k=1,2,\dots$, $0\le t\le1$,
determined by them, together with their discretizations given by
appropriate sequences of numbers $0=t_0^{(k)}<t_1^{(k)}<t_2^{(k)}
<\cdots<t_{s_k}^{(k)}=1$, satisfy the relations
$$ \allowdisplaybreaks
\align
d\(T_k'(\cdot),\bar T'_{k,t_0^{(k)},\dots,\,t_{s_k}^{(k)}}(\cdot)\)
&\Rightarrow0  \quad\text{if }k\to\infty, \tag3.8a \\
d\(T_k(\cdot)\,,\bar T_{k,t_0^{(k)},\dots,t_{s_k}^{(k)}}
(\cdot)\)&\Rightarrow0 \quad\text{if }k\to\infty,  \tag3.8b  \\
\sup_{0\le t\le 1}\left|\bar T_{k,t_0^{(k)},\dots,t_{s_k}^{(k)}}(t)
-\bar T'_{k,t^{(k)}_0,\dots,t^{(k)}_s}(t) \right|&\Rightarrow0
\quad\text{if }k\to\infty. \tag3.8c
\endalign
$$
 
By Theorem~A and relation (3.8a) the distributions of the stochastic
processes $\bar T'_{k,t_0^{(k)},\dots,\,t_{s_k}^{(k)}}(\cdot)$
converge to the distribution of the stochastic process
$T_0(\cdot)$ in the space $D([0,1])$. (Let us recall that the
distributions of the stochastic processes $T'_k(\cdot)$ and
$T_0(\cdot)$ agree.) Then  by relation~(3.8c) and Theorem~A
the distributions of the stochastic processes $\bar
T_{k,t_0^{(k)},\dots,t_{s_k}^{(k)}} (\cdot)$, and after this by
relation (3.8b) and Theorem~A the distributions of the stochastic
processes $T_k(\cdot)$ converge to the distribution of the stochastic
process $T_0(\cdot)$ in the space $D([0,1])$. Thus we have proved
the first part of the {\it Statement}.\/ After this, the second part of
the {\it Statement} follows from Theorem~A and the second part of
Proposition~3. Indeed, by this result
$$
d\(T_k(\cdot)\,,\bar T_{k,u_{k,0}^{(k)},\dots,u_{k,n_k}^{(k)}}
(\cdot)\)\Rightarrow0 \quad\text{if }k\to\infty.
$$
(We exploit in this step of the proof that the threshold index $k_0$
in formula (3.5) does not depend on the choice of the sufficiently
dense partition $0=t_0<t_1<\cdot<t_s$ of the interval~$[0,1]$.)
 
Let us remark that in Proposition~3 we have estimated the distance of
a stochastic process and its discretization in the metric
$d(\cdot,\cdot)$ introduced in the space $D([0,1])$ and not in the
supremum norm. We had to do so, since if the original stochastic
processes have jumps, and we have to work with such stochastic
processes, then these processes are far from their discretizations in
the supremum norm. On the other hand, if the points of jumps are not
too dense, which means that small intervals contain only at least one
jump, then under general conditions a stochastic process and its
sufficiently dense discretization are close to each other in the
metric $d(\cdot,\cdot)$. In the next Lemma~2 we give an estimate for
the $d(\cdot,\cdot)$ distance of two (simple) functions. It can help
us to estimate the $d(\cdot,\cdot)$ distance of stochastic processes.
In particular, it will be useful in the proof of Proposition~3.

\medskip\noindent
{\bf Lemma~2.} {\it Let $x(t)$ and $y(t)$, $0\le t\le1$, be two
cadlag functions on the interval $[0,1]$  with $p<\infty$ numbers 
of jumps. (We assume that two functions have the same number of 
jumps.) Let us also assume that the values of the functions 
$x(\cdot)$ and $y(\cdot)$ agree after the $j$-th jump, 
$0\le j\le p$. Let there exist a finite monotone sequence of 
numbers $0=t_0<t_1<\cdots<t_s=1$ such that $\inff_{1\le l\le s}
|t_l-t_{l-1}|\le\delta$ with some number $\delta>0$, and the
function $x(\cdot)$ is constant in all intervals $[t_{l-1},t_l)$,
$1\le l\le s-1$. Furthermore, we assume that if the $j$-th point of
jumps of the function $x(\cdot)$ is the point $t_{l_j}$ with some
$l_j\ge j$, then the $j$-th jump of the other function $y(\cdot)$
is in a point of the interval $(t_{l_{j-1}},t_{l_j}]$, $1\le j\le p$.
Then the inequality $d(x(\cdot),y(\cdot))<\delta$ holds.}
 
\medskip\noindent
{\it The proof of Lemma~2.}\/ Let $u_1,\dots, u_p$ be the points of
jumps of the function $y(\cdot)$. Let us define the following
homeomorphism $\lambda(\cdot)$ of the interval $[0,1]$ onto itself:
$\lambda (u_j)=t_{l_j}$, $1\le j\le p$, $\lambda(0)=0$, $\lambda(1)=1$,
and let the function $\lambda(\cdot)$ be linear in the intervals
$[u_{j-1},\,u_j]$, $1\le j \le p$, and $[0,u_1]$, $[u_p,1]$.
(The number $t_{l_j}$ is the $j$-th point of jump of the function
$x(\cdot)$.) Then $y(\lambda(\cdot))=x(\cdot)$, and $\supp_{0\le
t\le1}|\lambda(t)-t|\le \delta$. Hence $d(x(\cdot),y(\cdot))\le
\delta$, as we have claimed.
 
%\medskip\noindent
\vfill\eject

\noindent
{\script C.) The proof of Propositions~3 and~4.}

\medskip\noindent
{\it The proof of Proposition~3.}\/ Let us choose a number $\alpha>0$
such that the numbers $\pm\alpha$ are points of continuity of the
canonical measure $M_0$ on the real line, and $M_0(-\alpha,\alpha])
<\frac{\e\eta^2}8$, where $M_0(B)=N_0(B\times[0,1])$, $B\in\bold R^1$.
Let us also introduce the canonical measures $N_0''(\cdot)
=N_{0,\alpha}(\cdot)$ and $N'_{0,A_L}(\cdot)=N'_{0,A_L,\alpha}(\cdot)$
on the strip $\bold R^1\times[0,1]$ defined by the formulas
$N''_0(B)=N_0(B\cap \{(x,y)\colon\; |x|\ge \alpha\})$,
$N'_{0,A_L}(B)=N_0(B\cap \{(x,y)\colon\; A_L\le |x|<\alpha\})$ if 
$B\in \bold R^1\cap[0,1]$, and the numbers  $A_L$, $A_L>0$, are chosen 
in such a way that they can be applied in the regularized sums which 
define the stochastic process $T_0(t)$ by means of a Poissonian field. 
Let us also introduce the measures 
$\nu_0''(\,dx,\,dy)=\frac{N''_0(\,dx,\,dy)}{x^2}$ and
$\nu_{0,A_L}'(\,dx,\,dy)=\frac{N'_{0,A_L}(\,dx,\,dy)}{x^2}$ and consider 
a Poisson fields $X_n''=({X_n''}^{(1)}, {X_n''}^{(2)})$, $n=1,2,\dots$,
with counting measure $\nu''_0$ and the infinitely divisible field
$$
T_0''(t)=\summ_{n\colon\; {X_n''}^{(1)}>\alpha,\;0\le {X_n''}^{(2)}\le t}
{X_n''}^{(1)}-
E\(\sum_{n\colon\; {X_n''}^{(1)}>\alpha,\;0\le {X_n''}^{(2)}\le t}
\tau\({X_n''}^{(1)}\)\), \quad 0\le t\le 1,
$$
determined by it. Let us consider similarly a Poisson field
$X_{n,A_L}'=({X_{n,A_L}'}^{(1)},{X_{n,A_L}'}^{(2)})$, $n=1,2,\dots$
with counting measure $\nu'_{0,A_L}$ and the infinitely divisible
stochastic process
$$
T_{0,A_L}'(t)=
\summ_{n\colon\; A_L\le{X_n'}^{(1)}<\alpha,\; 0\le {X_n'}^{(2)}\le t}
{X_n'}^{(1)}-E\(
\summ_{n\colon\; A_L\le{X_n'}^{(1)}<\alpha,\; 0\le {X_n'}^{(2)}\le t}
\tau\({X_n'}^{(1)}\)\),
$$
$0\le t\le 1$, determined by this Poisson field. Put
$T_{0,A_L}(t)=T'_{0,A_L}(t)+T''_0(t)$, $0\le t\le1$. The stochastic
processes $T_{0,A_L}(\cdot)$ converge with probability~1 to the
stochastic process $T_0(\cdot)$ in the supremum norm. Hence to prove
formula (3.4) it is enough to show that
$$
P\(d(T_{0,A_L}(\cdot), \bar T_{0,A_L,t_0,\cdots,t_s}(\cdot))>\eta\)<\e
\quad \text{for all numbers } L\ge L_0,  \tag3.9
$$
where $L_0$  is an appropriate number, and $\bar
T_{0,A_L,t_0,\cdot,t_s}(\cdot))$ denotes the discretization of the
stochastic process $T_{0,A_L}(\cdot)$. Indeed,
$$
\left\{\oo\colon\; d(T_0,(\cdot,\oo), \bar T_0(\cdot,\oo))>\eta\right\}
\subset
\liminff_{L\to\infty}\left\{\oo\colon\;d(T_{0,A_L}(\cdot,\oo), \bar
T_{0,A_L,t_0,\cdots,t_s}(\cdot,\oo))>\eta\right\},
$$
hence formula (3.9) implies formula (3.4).
 
We can write
$$
\align
P\(d(T_{0,A_L}(\cdot),\bar T_{0,A_L,t_1,\dots,t_s}(\cdot))>\eta\)
&\le P\(\supp_{0\le t\le1}|T'_{0,A_L}(t)|>\frac\eta2\) \\
&\qquad +P\(d(T''_0(\cdot),\bar T''_{0,t_1,\dots,t_s}(\cdot))
>\frac\eta2\),
\endalign
$$
hence to prove formula (3.9) it is enough to show that
$$
P\(\supp_{0\le t\le1}|T'_{0,A_L}(t)|>\frac\eta2\)<\frac\e2, \quad
\text{if } L\ge L_0, \tag3.10
$$
and
$$
P\(d(T''_0(\cdot),\bar T''_{0,t_1,\dots,t_s}(\cdot))>\frac\eta2\)
<\frac\e2, \quad\text{if } \sup_{1\le l\le s}|t_l-t_{l-1}|<\delta
\tag3.11
$$
with some appropriate number  $\delta>0$.

As $T'_{0,A_L}(t)$ is a stochastic process with independent increments,
and its trajectories are cadlag functions hence we can write with the
help of the Kolmogorov inequality that
$$
P\(\supp_{0\le t\le1}|T'_{0,A_L}(t)|>\frac\eta2\)\le
\frac{4ET'_{0,A_L}(1)^2}{\eta^2}=\frac4{\eta^2}\int
u^2\mu'_{0,A_L}(\,du)\le\frac4{\eta^2} M_0([-\alpha,\alpha])<\e,
$$
where the measure $\mu'_{0,A_L}$ is defined by the relation
$\mu'_{0,A_L}(B)=\nu'_{0,A_L}(B\times [0,1])$ for all measurable 
sets $B\subset \bold R^1$. Hence inequality (3.10) holds.
 
To prove formula (3.11) let us first introduce the function
$\lambda_0(t)=\lambda_{0,\alpha}(t)$ defined by the formula
$\lambda_0(t)=\nu''_0(\bold
R^1\times[0,t])=\int_{\{(x,y)\colon\; |x|>\alpha,\; 0\le|y|<t\}}
\frac{N_0(\,dx,\,dy)}{x^2}$, $0\le t\le1$.
Let us remark that because of Condition~b.) of Theorem~2
the function $\lambda_0(\cdot)$  is continuous in the interval $[0,1]$.
We claim that thee exists some number
$\delta>0$ such that for all partitions
$0=t_0<t_1<\cdots<t_s=1$ of the interval $[0,1]$
$$
\summ_{l=1}^s \(\lambda_0(t_l)-\lambda_0(t_{l-1})\)^2<\e
\quad \text{if }|t_l-t_{l-1}|<\delta\text { for all numbers } 1\le
l\le s. \tag3.12
$$
Indeed, as the function $\lambda_0(\cdot)$ is uniformly continuous,
hence there exists a number $\delta>0$ such that
$|\lambda_0(t)-\lambda_0(s)|<\frac{\e}{\lambda(1)}$ if
$|t-s|<\delta$. Furthermore, the function $\lambda_0(\cdot)$ is
monotone increasing. Hence $\summ_{l=1}^s
\(\lambda_0(t_l)-\lambda_0(t_{l-1})\)^2\le \supp_{1\le l\le s}
|\lambda_0(t_l)-\lambda_0(t_{l-1})|\summ_{l=1}^s
|\lambda_0(t_l)-\lambda_0(t_{l-1})|<\e$ if $|t_l-t_{l-1}|<\delta$
for all numbers $1\le l\le s$, i.e.\ formula (3.12) is valid.
 
A Poisson distributed random variable with parameter $\lambda$ takes a
value more than or equal to two with probability $1-e^{-\lambda}-\lambda
e^{-\lambda}\le\frac{ \lambda^2}2$. Hence the probability of the event
that a Poisson field $X_n''=({X_n''}^{(1)}, {X_n''}^{(2)})$,
$n=1,2,\dots$, on the strip $\bold R^1\times[0,1]$ with counting
measure $\nu''_0$ contains at least two such points
${X_{n_1}''}$ and ${X_{n_2}''}$ whose second coordinates
${X''_{n_1}}^{(2)}$ and ${X''_{n_2}}^{(2)}$ are in an interval $[s,t]$,
$0\le s<t\le1$, is less than $\frac12\nu''(\bold R^1\times[s,t])^2
=\frac12(\lambda(t)-\lambda(s))^2$. Hence by formula (3.12)
for a Poisson field $X_n''=({X_n''}^{(1)}, {X_n''}^{(2)})$,
$n=1,2,\dots$, with counting measure $\nu''_0$ and a partition
$0\le t_0< t_1<\cdots<t_s=1$ of the interval $[0,1]$ the probability
of the event
$$
A(t_1,\dots,t_s)=\{\oo\colon\;\#\{n\colon\;t_{l-1}\le {X_n''}^{(2)}(\oo)
\le t_l\}\le1 \quad \text{for all numbers } 1\le l\le s\}
$$
can be estimated as
$$
1-P\(A(t_1,\dots,t_s)\)=P\(\Omega\setminus A(t_1,\dots,t_s)\)
\le\frac12\sum_{l=1}^s\(\lambda_0(t_l)-\lambda_0(t_{l-1})\)^2
\le \frac\e2 \tag3.13
$$
if $\supp_{1\le l\le s}|t_l-t_{l-1}|<\delta$ with a sufficiently small
$\delta>0$.
 
Let us fix a number $\delta>0$ with which formula (3.13) holds together
with the inequality $\delta<\frac\eta2$. Then formula (3.13) means that
on a set of probability greater than $1-\frac\e2$ the cadlag functions
$x(t)=\bar T''_{0,t_0,t_1,\dots,t_s}(t,\oo)$ and $y(t)=T_0''(t,\oo)$
together with the partition $0=t_0<t_1<\cdots<t_s=1$ of the interval
$[0,1]$ satisfy the conditions of Lemma~2, and $d(T''_0(\cdot,\oo),
\bar T''_{0,t_0, \cdots,t_s}(\cdot,\oo))\le\delta<\frac\eta2$ on a set
of probability greater than $1-\frac\e2$. This fact implies relation
(3.11) and as a consequence also relation (3.4).
 
To prove formula (3.5) it is enough to prove the following analogs of
formulas (3.10) and (3.11).
$$
P\(\supp_{0\le t\le1}|T'_{k,A_L}(t)|>\frac\eta2\)<\frac\e2, \quad
\text{if } k\ge k_0\text{ and } L\ge L_0 \tag3.14
$$
with an appropriate threshold index $k_0=k_0(\e, \eta)$, and
$$
P\(d(T''_k(\cdot),\bar T''_{k,t_1,\dots,t_s}(\cdot))>\frac\eta2\)
<\frac\e2, \quad\text{if }  k\ge k_0,\text{ and } \sup_{1\le l\le
s}|t_l-t_{l-1}|<\delta \tag3.15
$$
with an appropriate threshold index $k_0=k_0(\e,\eta)$ and number
$\delta>0$.

Formula (3.14) can be proved similarly to formula (3.10). The only
difference is that now we exploit that since the numbers
$\pm\alpha$ are points of continuity of the measure $M_0$, and
$M_0([-\alpha,\alpha])<\frac{\e\eta^2}8$ hence also the relation
$M_k(-\alpha,\alpha)<\frac{\e\eta^2}8$ holds if $k\ge k_0$ with an
appropriate threshold index $k_0$. (We define the measures $M_k$,
$k=1,2,\dots$, on the real line, analogously to the definition of the
measure $M_0$, by the formula $M_k(B)=N_k(B\times[0,1])$ for all
measurable sets $B\in\bold R^1$.)
 
Formula (3.15) can be proved similarly to formula (3.11). The only
difference is that now we have to prove and apply the following analog
of relation (3.12).
 
Let us introduce the functions $\lambda_k(t)=\lambda_{k,\alpha}(t)$
defined by the formula $\lambda_k(t)=\nu''_k(\bold R^1\times[0,t])
=\int_{\{(x,y)\colon\; |x|>\alpha,\; 0\le|y|<t\}}
\frac{N_k(\,dx,\,dy)}{x^2}$, $0\le t\le1$ for all indices
$k=1,2,\dots$. Then there exists number
$\delta>0$ and threshold index $k_0=k_0(\delta)$ in such a way that for
all partitions $0=t_0<t_1<\cdots<t_s=1$ of the interval $[0.1]$
$$
\summ_{l=1}^s \(\lambda_k(t_l)-\lambda_k(t_{l-1})\)^2\le \e
\quad \text{if } k\ge k_0\text{ and } |t_l-t_{l-1}|<\delta \text
{ for all numbers } 1\le l\le s. \tag3.16
$$
Let us emphasize the threshold index $k_0$ in formula (3.16)
depends only on the number $\delta>0$ and not on the partition
$0=t_0<t_1<\cdots<t_s=1$ of the interval $[0,1]$.
 
Let us prove formula (3.16) first in the special case if the partition
$0=t_0<t_1<\cdots<t_s=1$ of the interval $[0,1]$ satisfies not only the
inequality $|t_l-t_{l-1}|<\delta$, but also the inequality
$|t_l-t_{l-1}|\ge\frac\delta2$ for all numbers $1\le l\le s$. The
convergence of the canonical measures $N_k$ to the canonical measures
$N_0$ and the continuity of the measure $\lambda_0(\cdot)$ imply that
the monotone functions $\lambda_k(\cdot)$ converge in all points of the
interval $[0,1]$ to the monotone and continuous function
$\lambda_0(\cdot)$. The properties of the functions $\lambda_k(\cdot)$
also imply that the convergence
$\limm_{k\to\infty}\lambda_k(t)=\lambda_0(t)$, $0\le t\le1$, is
uniform. Beside this, the sum at the right-hand side of formula (3.16)
contains at most $\frac2\delta$ terms. Hence formula (3.12) implies
formula (3.16) with the same number $\delta$ and a sufficiently large
threshold index~$k_0(\delta)$.
 
Let us now consider a partition $0=t_0<t_1<\cdots<t_s=1$ of the
interval such that $\supp_{1\le s\le k}|t_l-t_{l-1}|<\frac\delta2$
where formula (3.16) holds in the special case considered in the
previous paragraph with the number $\delta$. It is not difficult to
see that the sequence of numbers $0=t_0<t_1<\cdots<t_s=1$ has a
subsequence $0=t_{j_0}<t_{j_1}<\cdots<t_{j_p}=1$ such that
$\frac\delta2\le \left|t_{j_u}-t_{j_{u-1}}\right|<\delta$ for all
numbers $1\le u\le p$. Let us take such a subsequence. Then we can
write with the help of the already proven case that
$$
\summ_{l=1}^s \(\lambda_k(t_l)-\lambda_k(t_{l-1})\)^2\le
\summ_{u=1}^p \(\lambda_k(t_{j_u})-\lambda_k(t_{j_{u-1}})\)^2< {\e}
$$
if $k\ge k_0(\delta)$. In such a way we have proved formula (3.16)
(with the choice $\frac\delta2$ instead of the number $\delta$.) After
the proof of formulas (3.15) and (3.16) formula (3.5) can be proved in
the same way as formula (3.4).
 
\medskip\noindent
{\it The proof of Proposition~4.}\/ The proof of Proposition~4 is based
on the following observation. To determine the difference
$\bar T_k(\cdot)-\bar T_k'(\cdot)$ of the discretizations of the
processes $T_k(\cdot)$ and $T_k'(\cdot)$ we need not know the
precise values of the Poisson fields $X_{k,n}
=(X_{k,n}^{(1)},X_{k,n}^{(2)})$ and $X'_{k,n}
=({X'_{k,n}}^{(1)},{X'_{k,n}}^{(2)})$, $n=1,2,\dots$, $k=1,2,\dots$,
which determine these processes. The knowledge of the values of the
second coordinates of these fields is not necessary, it is enough to
know in which one of the intervals $[t_{l-1},t_l]$ they lie.
Hence in the first step of the construction we do not decide the
precise value of the Poisson fields we have to define. In such a way in
the first step of the construction a coupling problem has to be handled
which can be solved relatively simply with the help of Proposition~2.
Then the construction can be completed by an appropriate randomization.
 
To work out the details first we introduce some notations. Let us define
the canonical measures $\tilde N_{k,l}$, $k=0,1,2,\dots$, $1\le l\le s$,
on the real line by the formula $\tilde N_{k,l}(B)
=N_k(B\times(t_{l-1},t_l])$, $k=0,1,2,\dots$, $1\le l\le s$. Let us
then define the canonical measures $\tilde N'_{k,l}$ on the strip
$\bold R^1\times [0,1]$ by the formulas $\tilde N'_{k,l}(B\times
\{t_l\})=\tilde N'_{k,l}(B)$, $B\subset \bold R^1$, $k=0,1,2,\dots$,
$1\le l\le s$, i.e. let the measure $\tilde N'_{k,l}$ be the shift of
the measure $\tilde N_{k,l}$ from the real line to the line $\{(x,y)\colon\;
y=t_l\}$ parallel to the coordinate axis in $\bold R^2$. Let us also
define the canonical measures $\tilde N_k$, $k=0,1,2,\dots$, on the
strip $\bold R^1\times [0,1]$ as $\tilde N_k=\summ_{l=1}^s \tilde
N'_{k,l}$, $k=0,1,2,\dots$.
 
Let us define the measures $\tilde\nu_{k,l}(dx)=\frac{\tilde
N_{k,l}(dx)}{x^2}$, $\tilde\nu'_{k,l}(dx)=\frac{\tilde
N'_{k,l}(dx)}{x^2}$ and $\tilde\nu_{k}(dx,dy)=\frac{\tilde
N_k(dx,dy)}{x^2}$, $k=0,1,2,\dots$, $l=1,2,\dots,s$.
By Proposition~2 some Poisson fields $\xi_{k,l}(n)$, $n=1,2,\dots$,
$k=1,2,\dots$, $l\le 1\le s$, can be constructed with counting measures
$\tilde\nu_{k,l}$ together with some other Poisson fields
$\xi'_{k,l}(n)$, $n=1,2,\dots$, $k=1,2,\dots$, $1\le l\le s$, with
counting measures $\tilde\nu_{0,l}$ in such a way that the random
variables $U_{k,l}$ and $U'_{k,l}$ with infinitely divisible
distributions determined by these Poisson fields satisfy the relations
$U'_{k,l}-U_{k,l}\Rightarrow0$ as $k\to\infty$ for all numbers
$1\le l\le s$, where $\Rightarrow$ denotes stochastic convergence.
Let us also define the random variables $\tilde T_{k,j}=\summ_{l=1}^j
U_{k,l}$, $\tilde T'_{k,j}=\summ_{l=1}^j U'_{k,l}$, $k=1,2,\dots$,
$1\le j\le s$. Then also the relation $\supp_{1\le j\le s}\left|\tilde
T_{k,j}-\tilde T'_{k,j}\right|\Rightarrow0$ holds as $k\to\infty$.
 
Now we begin the construction of Poisson fields for which the
discretizations of the stochastic processes $T_k(t)$ and $T'_k(t)$
determined by them satisfy formula~(3.6). Let us consider the measures
$\nu_{k,l}$, $k=0,1,2,\dots$, $1\le l\le s$, which are restrictions of
the measures $\nu_k(dx,dy)=\frac{N_k(dx,dy)}{x^2}$ to the strip $\bold
R^1\times (t_{l-1},t_l]$. Then for all integers $k=0,1,2\dots$, $1\le
l\le s$, points $x\in\bold R^1$, and measurable sets $B\in\bold
R^1\times (t_{l-1},t_l]$ there exist such ``conditional measures"
$\nu_{k,l}(B|x)$ on the interval $(t_{l-1},t_l]$ for which
$\nu_{k,l}(\,\cdot\,|x)$ is a probability measure in the strip $\bold
R^1\times(t_{l-1},t_l]$ for all numbers $x\in\bold R^1$,
$\nu_{k,l}(B|\,\cdot\,)$ is a measurable function on the real
line for all measurable sets $B\subset\bold R^1\times(t_{l-1},t_l]$,
and
$$
\nu_{k,l}(B)=\int \nu_{k,l}(B|x)\tilde\nu'_{k,l}(\,dx)\quad \text{for
all measurable sets } B\subset \bold R^1\times(t_{l-1},t_l] \tag3.17
$$
for all numbers $k=0,1,2,\dots$ and $1\le l\le s$, where
$\tilde\nu_{k,l}$ is the measure defined in the previous paragraph.
The existence of a ``conditional measure"
$\nu_{k,l}(\,\cdot\,|\,\cdot\,)$ satisfying relation (3.17) is a
consequence of a classical result of probability theory about the
existence of regular conditional distributions.
 
Now we construct Poisson fields $X_{k,n}
=(X_{k,n}^{(1)},X_{k,n}^{(2)})$, $n=1,2,\dots$, $k=1,2,\dots$
with counting measures $\nu_k$ and Poisson fields $X'_{k,n}
=({X'_{k,n}}^{(1)},{X'_{k,n}}^{(2)})$, $n=1,2,\dots$, $k=1,2,\dots$
with counting measures $\nu_0$ such that the stochastic processes
$T_k(t)$ and $T'_k(t)$ determined by them satisfy Proposition~4.
Let us consider the already constructed Poisson fields $\xi_{k,l}(n)$,
$n=1,2,\dots$, $k=1,2,\dots$, $l\le 1\le s$, with counting measures
$\tilde\nu_{k,l}$ and the Poisson fields $\xi'_{k,l}(n)$, $n=1,2,\dots$,
$k=1,2,\dots$, $1\le l\le s$, with counting measures $\tilde\nu_{0,l}$.
For all points $\xi_{k,l}(n)$ and $\xi'_{k,l}(n)$ of these Poisson
fields we shall construct random variables $\eta_{k,l}(n)$ and
$\eta'_{k,l}(n)$ in a random way, and the Poisson fields we want to
construct will consist of the points $(\xi_{k,l}(n),\eta_{k,l}(n))$
and $(\xi'_{k,l}(n),\eta'_{k,l}(n))$. Let us construct for all random
variables $\xi_{k,l}(n)$ a $\nu_{k,l}(\,\cdot\,|\xi_{k,l}(n))$
distributed random variable $\eta_{k,l}(n)$ and for all random
variables $\xi_{k,l}'(n)$ a $\nu_{0.l}(\,\cdot\,|\xi'_{k,l}(n))$
distributed random variables $\eta'_{k,l}(n)$ on the interval
$[t_{l-1},t_l]$. Let us construct these random numbers independently of
each other. For a fixed index~$k$ let the Poison field $X_{k,n}=
\(X_{k,n}^{(1)},X_{k,n}^{(2)}\)$, $n=1,2,\dots$, $k=1,2,\dots$ with
counting measure $\nu_k$ consist of the previously constructed pairs
of points $(\xi_{k,l}(n),\eta_{k,l}(n))$, $n=1,2,\dots$, and similarly
let the Poisson field $X'_{k,n}=\({X'_{k,n}}^{(1)},{X'_{k,n}}^{(2)}\)$,
$n=1,2,\dots$, $k=1,2,\dots$ with counting measure $\nu_0$ consist of
the pairs of points $(\xi'_{k,l}(n),\eta'_{k,l}(n))$, $n=1,2,\dots$,
$1\le l\le s$. We claim that the above constructed Poisson fields
satisfy Proposition~4.
 
We shall prove that if the rectangle $B\times [t_{l-1},t_l]$ satisfies
the property $\nu_{k,l}(B\times [t_{l-1},t_l])<\infty$, then the points
falling to this rectangle $B\times [t_{l-1},t_l]$ define a Poisson
field with counting measure $\nu_{k,l}$ on this rectangle. We shall
show this for all numbers $k=1,2,\dots$ and $1\le l\le s$. Beside
this we claim that if the property $\nu_{0,l}(B\times [t_{l-1},t_l])
<\infty$ holds, then the points $(\xi'_{k,l}(n),\eta'_{k,l}(n))$,
$n=1,2,\dots$, falling to the rectangle $B\times [t_{l-1},t_l]$ define
a Poisson field with counting measure $\nu_{0,l}$ on this rectangle for
all numbers $k=1,2,\dots$ and $1\le l\le s$. These facts imply that we
have really constructed Poisson fields with the right counting measure.
These statements can be simply proved with the help of the following
observation. The distributions of these point processes agree with the
distributions of the point process we get in the following way: Let us
choose randomly many number of points with Poisson distribution with
parameter $\nu_{k,l}(B\times [t_{l-1},t_l])$ and drop them randomly to
the rectangle $B\times [t_{l-1},t_l]$ independently of each other with
distribution $\frac{\mu_{k,l}(dx,dy)}{\nu_{k,l}(B\times
[t_{l-1},t_l])}$. Such constructions supply Poisson fields with the
right counting measure.
 
Finally we remark that the above constructed Poisson fields are such
that the discretizations of the infinitely divisible stochastic
processes $T_k(\cdot)$ and $T_k'(\cdot)$ determined by them satisfy
the inequalities $\bar T_{k,t_0,\cdots,t_s}(t)=\tilde T_{k,j-1}$  and
$\bar T'_{k,t_0,\cdots,t_s}(t)=\tilde T_{k,j-1}'$ if $t_{j-1}<t \le
t_j$, $1\le j\le s$, and $\bar T_{k,t_0,\cdots,t_s}(1)=\tilde T_{k,s}$,
$\bar T'_{k,t_0,\cdots,t_s}(1)=\tilde T'_{k,s}$, $k=1,2,\dots$,
Hence the processes $\bar T_{k,t_0,\cdots,t_s}(\cdot)$ and $\bar
T'_{k,t_0,\cdots,t_s}(\cdot)$ satisfy formula (3.7). Proposition~4 is
proved.
 
\beginsection Appendix. The proof of Theorem A.
 
{\it The proof of Theorem A.} It is enough to prove the statement
formulated in general separable metric spaces. The weak convergence of
the random variables $S_k$ to a probability measure $\mu$  can be
formulated so that the distributions $\mu_k$ of the random variables
$S_k$ satisfy the relation $\limsupp_{k\to\infty}\mu_k(\bold F)\le
\mu(\bold F)$ for all closed sets $\bold F\subset X$. We shall show
that under the conditions of Theorem~A the distributions $\bar\mu_k$ of
the random variables $T_k$ also satisfy the relation
$\limsupp_{k\to\infty}\bar\mu_k(\bold F)\le\mu(\bold F)$ for all closed
sets $F\subset X$. (Let us remark that the characterization of the
weak convergence applied in this proof is valid in all separable metric
spaces. We do not have to assume that the metric space is complete.
(See for instance Theorem~2.1 in the book of
P.~Billingsley {\it Convergence of Probability Measures.}\/)
 
 
Let us fix some number $\e>0$. As $\bold F=\bigcapp_{n=1}^\infty \bold
F_{\frac1n}$, where $\bold F_a=\{x\colon\;\rho(x,\bold F)\le a\}$, hence
there exists a number $\delta=\delta(\e)>0$ such that $\mu(\bold F)\ge
\mu(\bold F_\delta)-\e$. Furthermore, the inequality $\mu_k(\bold
F_\delta)<\mu(\bold F_\delta)+\e$ holds if $k\ge k_0=k_0(\e,\delta,
\bold F)$. As $\rho(S_k,T_k)$ tends to zero stochastically, hence
also the inequality $\bar\mu_k(\bold F)=P(T_k\in \bold F)\le P(S_k\in
\bold F_\delta)+P(\rho(S_k,T_k)>\delta)\le \mu_k(\bold F_\delta)+\e$,
holds if $k\ge k_0$ and the threshold index $k_0=k_0(\e,\delta, \bold
F)$ is chosen sufficiently large. The above inequalities imply that
$\bar\mu_k(\bold F)\le \mu_k(\bold F_\delta)+\e\le \mu(\bold
F_\delta)+2\e\le \mu(\bold F)+3\e$ if $k\ge k_0(\e,\delta,\bold F)$.
The above inequality holds for all numbers $\e>0$, and this implies
that $\limsupp_{k\to\infty}\bar\mu_k(\bold F)\le\mu(\bold F)$. Thus
Theorem~A is proved.
 
 
 
 
 
 
\bye