\magnification=\magstep1
\hsize=16truecm
 
\input amstex
\TagsOnRight
\parindent=20pt
\parskip=2pt plus 1pt
\define\A{{\bold A}}
\define\BB{{\bold B}}
\define\DD{{\bold D}}
\define\T{{\bold T}}
\define\U{{\bold U}}
\define\({\left(}
\define\){\right)}
\define\[{\left[}
\define\]{\right]}
\define\e{\varepsilon}
\define\oo{\omega}
\define\const{\text{\rm const.}\,}
\define\supp {\sup\limits}
\define\inff{\inf\limits}
\define\summ{\sum\limits}
\define\prodd{\prod\limits}
\define\limm{\lim\limits}
\define\limsupp{\limsup\limits}
\define\liminff{\liminf\limits}
\define\bigcapp{\bigcap\limits}
\define\bigcupp{\bigcup\limits}
\def\Re{\text{\rm Re}\,}
\def\Im{\text{\rm Im}\,}
\font\script =cmcsc10
 
  
\centerline{{\bf Limit theorems and infinitely divisible
distributions.} \ \  {\rm Part II}}
\smallskip
\centerline{\it by P\'eter Major}
\centerline{The Mathematical Institute of the Hungarian Academy of
Sciences} 

\medskip\noindent
{\narrower {\narrower
{\it Summary:}\/ In the second part of this work we deal with the
question when the normalized partial sums of independent random
variables, or more generally the sums of the random variables in
the same row of a triangular array converge in distribution.
We present a result which gives a necessary and sufficient condition
for the existence of a limit distribution if the sequence of random
variables or the triangular array satisfies the uniform smallness
condition, and also describe the limit distribution. It turns out that
the limit is always an infinitely divisible distribution. The hard part
of the problem is to show that the sufficient condition given for the
existence of a limit distribution is at the same time a necessary
condition. We discuss the content of this condition in more detail and
also show how the most important classical limit theorems can be
obtained as special cases of the result discussed in this work.
 
We also try to explain the main ideas of the proof. An important step
in it is the introduction of the so-called associated distributions
of the distribution functions of the summands and to show that the
convergence of independent random variables with these associated
distributions is closely related to the original limit problem. The
associate distributions are infinitely divisible. So to understand
limit theorems for sums of independent random variables it is useful
to study the special problem when the sums of independent random
variables with infinitely divisible distributions have a limit.
\par}\par}
 
\medskip\noindent
{\bf 1. Formulation of the basic results.}

\medskip\noindent
We present the proof of a result which gives a necessary and
sufficient condition for the existence of a limit distribution for the
(normalized) sums of the random variables in the same row of a
triangular array if they satisfy the uniform smallness condition.
Furthermore, the limit distribution will be also described. It turns
out that it is always an infinitely divisible distribution. This means
that if the uniform smallness condition holds, then the limit
distribution of the normalized sums of independent random variables is
infinitely divisible in the most general case. Let us emphasize that we
did not assume that the summands are identically distributed.
 
To study the limit problem we are interested in it is useful to
associate to all terms in the triangular array we are working with 
an infinitely divisible random variable in an appropriate way, We 
associate infinitely divisible and independent random variables to 
the random variables in the same row. It can be achieved that the 
sums of these associated random variables from a row of the 
triangular array converge in distribution if and only if the 
original sums we are investigating converge. Moreover, the original 
sums and the sums made from the associated random variables have 
the same limit distribution. Since the sum of independent and
infinitely divisible random variables is again infinitely 
divisible, the introduction of these associated random variables 
leads to the problem when a sequence of infinitely divisible 
distributions have a limit.
 
\headline{\ifodd\pageno \hfill {\it Limit theorems and infinitely
divisible distributions.} {\rm Part II} \hfill \else
\hfill {\it P\'eter Major} \hfill \fi}
 
Naturally, the results proved for general triangular arrays also hold
in the special case when the elements in a row are not only independent
but also identically distributed random variables. This particular case
deserves special attention, because in this case the following
heuristically ``obvious" statement holds which nevertheless demands a
special proof: If the sums of the random variables from distinct rows
have a limit distribution and the random variables in a row of this
triangular array are not only independent but also identically
distributed, then the triangular array satisfies the uniform smallness
condition. This result will be exploited in the proof of the
L\'evy--Hinchin formula i.e.\ in the proof of the result that the
construction in Part~I describes all possible infinitely
divisible distributions, and it describes them in a unique way.
 
Before formulating the results we are interested in let us recall some
important notions and let us introduce some notations. Let $\xi_{k,j}$,
$k=1,2,\dots$, $j=1,\dots,n_k$, be a triangular array of random
variables, i.e.\ let us assume that for a fixed number $k$ the random
variables $\xi_{k,j}$, $1\le j\le n_k$, are independent. This
triangular array satisfies the condition of uniform smallness if for
all numbers $\e>0$ $\limm_{k\to\infty}\supp_{1\le j\le
n_k}P(|\xi_{k,j}|>\e)=0$. Let us also recall the notion of canonical
measures introduced in Part~I. A measure $M$ on the real line is
called a canonical measure if for all finite intervals
$[a,b]\subset\bold R^1$ the measure $M\{[a,b]\}$ is finite, and for an
arbitrary number $a>0$
$$
\int_{a}^\infty\frac1{x^2}M(\,dx)<\infty,\quad \text{and}\quad
\int_{-\infty}^{-a}\frac1{x^2}M(\,dx)<\infty.
$$
 
Let $\xi_{k,j}$, $k=1,2,\dots$, $j=1,\dots,n_k$, be a triangular array
satisfying the uniform smallness condition, and let $F_{k,j}$  denote
the distribution of the random variable $\xi_{k,j}$. Let us introduce
the $\sigma$-finite measures
$$
M_k(\,dx)=\sum_{j=1}^{n_k}x^2 F_{k,j}(\,dx) \tag1.1
$$
and the functions
$$
\aligned
M^+_k(x)&=\sum_{j=1}^{n_k}(1-F_{k,j}(x))=\int_{u\ge x}
\frac1{u^2}M_k(\,du), \\
M^-_k(x)&=\sum_{j=1}^{n_k}F_{k,j}(-x)
=\int_{u<-x} \frac1{u^2}M_k(\,du),
\endaligned
\qquad k=1,2,\dots,\;\; x>0 \tag1.2
$$
on the real line for all $k=1,2,\dots$. Let us introduce the random sums
$S_k=\summ_{j=1}^{n_k}\xi_{k,j}$. The theorem formulated below gives a
necessary and sufficient condition for the convergence in distribution
of the normalized sums $S_k-b_k$ with appropriate norming constants
$b_k$. This condition is expressed by means of the above introduced
measures $M_k$ and functions $M_k^\pm$. 

\medskip\noindent
{\bf  Theorem 1.} {\it Let $\xi_{k,j}$, $k=1,2,\dots$, $j=1,\dots,n_k$,
be a triangular array satisfying the uniform smallness condition. Let
$F_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, denote the distribution
function of the random variable $\xi_{k,j}$ and set
$S_k=\summ_{j=1}^{n_k}\xi_{k,j}$. Let us assume that the normalized
sums $S_k-\bar b_k$ with some appropriate constants $\bar b_k$ converge
in distribution as $k\to \infty$. Then there exists a canonical measure
$M$ on the real line such that the functions $M_k^\pm(x)$ defined
in formula (1.2) together with the functions
$$
M^+(x)=\int_{u\ge x}\frac1{u^2}M(\,du), \qquad
M^-(x)=\int_{u<-x}\frac1{u^2}M(\,du)
\tag$1.2'$
$$
defined by means of the canonical measure $M$ in an analogous way
satisfy the relation
$$
\lim_{k\to\infty}M_k^+(x)=M^+(x) \quad \text {and} \quad
\lim_{k\to\infty}M_k^-(x)=M^-(x)
\tag1.3
$$
in all such points~$x>0$ where the functions $M^+(\cdot)$ or
$M^-(\cdot)$ are continuous. Besides, the relation (1.6)
formulated below also holds. To formulate this relation let us
first introduce some notations.
 
Let us fix a number $a>0$, and define the function
$$
\tau(x)=\tau_a(x)=\cases
x&\text{if }|x|\le a\\
a&\text{if }x\ge a  \\
-a&\text{if }x\le -a
\endcases  \tag1.4
$$
and numbers
$$
\align
\beta_{k,j}=\beta_{k,j}(a)=E\tau(\xi_{k,j}),\quad
b_k&=b_k(a)=\sum_{j=1}^{n_k}\beta_{k,j},
\quad B_k=B_k(a)=\sum_{j=1}^{n_k}\beta^2_{k,j},\\
&\qquad k=1,2,\dots,\quad j=1,\dots,n_k. \tag1.5
\endalign
$$
If the normalized sums $S_k-\bar b_k$ converge in distribution with
some appropriate constants $\bar b_k$, then the normalized sums
$S_k-b_k$ with the above introduced numbers~$b_k$ also converge in
distribution, and the canonical measures $M_k$ introduced in formula
(1.1) satisfy the relation
$$
M_k([-s,s])-B_k\to M([-s,s]), \quad\text{if }k\to\infty \tag1.6
$$
in all points $s>0$ which are points of continuity of $M^+$ and 
$M^-$. Here we worked with the previously introduced canonical 
measure~$M$ and functions~$M^\pm(\cdot)$.
 
Conversely, if the relations (1.3) and (1.6) hold with an appropriate
canonical measure $M$ and the numbers $B_k$ defined in formula (1.5),
then the appropriate normalizations of the sums $S_k$ converge in
distribution. More explicitly, in this case the normalized sums
$S_k-b_k$, $k=1,2,\dots$, with the constants $b_k$ defined in formula
(1.5) have a limit distribution which we can describe. The
characteristic function $\varphi(t)$ of this limit distribution has
a logarithm which can be given by the formula
$$
\log \varphi(t)=\int_{-\infty}^\infty \frac{e^{itu}-1-it\tau(u)}{u^2}
M(\,du) \tag1.7
$$
with the canonical measure $M$ determined by formulas (1.3), $(1.2')$
and (1.6) and the function $\tau$ defined in formula (1.4). (Formula
(1.6) is needed to define the measure $M(\{0\})$ of the origin.)
 
To satisfy the above limit theorem it is sufficient to demand a weakened
version of condition (1.6), namely to demand that it hold in a point of
(joint) continuity of the functions $M^\pm$. (See Remark~1 formulated
below.)}

\medskip
To give a complete formulation of Theorem~1 or more explicitly of
formula (1.7) in it we still have to define the value of the integrand
in (1.7) also in the point $u=0$. By continuity arguments this is
defined by the relation
$$
\left.\frac{e^{itu}-1-it\tau(u)}{u^2}\right|_{u=0}=
\lim_{u\to0}\frac{e^{itu}-1-it\tau(u)}{u^2}=-\frac{t^2}2
$$
here and in subsequent formulas.
 
Let us consider a triangular array $\xi_{k,j}$, $k=1,2,\dots$,  
$1\le j\le n_k$, satisfying the uniform smallness  condition. 
Theorem~1 gives a necessary and sufficient condition for the 
existence of an appropriate normalizations $S_k-b_k$ of the sums
$S_k=\summ_{j=1}^{n_k}\xi_{k,j}$  for which these normalized sums
converge in distribution. Besides, this result also gives a
possible normalization and describes the limit distribution belonging
to it. This limit distribution is described in formula (1.7) if the
norming constants $b_k$ are defined in~(1.5). A
comparison of this formula with the results of Part~I of this work
yields that this limit is an infinitely divisible distribution whose
Poissonian component is the (regularized) sum of the values of a
Poisson process with counting measure $ u^{-2}M(du)$ and which has a
Gaussian component with expectation zero and variance $M(\{0\})$.
The measure $M$ appearing here is the limit of the measures
$M_k$ introduced in formula (1.1). The limit procedure leading to
this measure $M$ is described through formulas (1.3), (1.5) and (1.6).
Let us emphasize that the limit in Theorem~1 is always an infinitely
divisible distribution, although we did not assume that the terms in
the sums we have considered are identically distributed. (In the first
part of this work we only gave a heuristic argument why the limit of
the sums of independent and identically distributed random variables
have always an infinitely divisible distribution.) A result presented
in the later formulated Theorem~$2'$ also implies that the limit
distribution determines the measure~$M$ in formula (1.7).
 
To understand Theorem~1 better we make some comments. Let us first
observe that if the sequence of the random variables $S_k-b_k$ converges
in distribution, then the sequence $S_k-\bar b_k$  with another sequence
of constants $\bar b_k$ converges in distribution if and only if the
finite limit $C=\limm_{k\to\infty}(\bar b_k-b_k)$ exists. Indeed, if the
above finite limit exists, and the sequence $S_k-b_k$ converges to a
distribution $F(x)$, then the sequence $S_k-\bar b_k$ converges to the
distribution $F(x+C)$. Conversely, if the sequence $\bar b_k-b_k$ is
non-convergent, then either this sequence is non-bounded and it has a
subsequence $\bar b_{k_j}-b_{k_j}$ which tends to plus or minus
infinity and the measures of all compact sets tends to zero with respect
to the distributions of the subsequences $S_{k_j}-\bar b_{k_j}$ in this
case, or this sequence has two subsequences with indices $k_j$ and $\bar
k_j$ such that the subsequences $\bar b_{k_j}-b_{k_j}$ and $\bar b_{\bar
k_j}-b_{\bar k_j}$ have two different finite limits. In the latter case
the sequence $S_k-\bar b_k$ has two subsequences with different limits.

The above observation tells us how many freedom we have in the choice of
the norming constant $b_k$ in Theorem~1. In its formulation we fixed a
parameter $a>0$ and the constants $b_k$ depended on this number~$a$
through the function $\tau(\cdot)=\tau_a(\cdot)$. If $a$ and $a'$ are
two different positive constants, then
$$
b_k(a)-b_k(a')=\int\frac{(\tau_a(u)-\tau_{a'}(u))}{u^2}M_k(\,du),
$$
and the finite limit $C=\limm_{k\to\infty}\(b_k(a)-b_k(a')\)
=\int\frac{(\tau_a(u)-\tau_{a'}(u))}{u^2}M(\,du)$ exists by
formula (1.3). Besides, the normalizations $b_k(a)$ and $b_k(a')$
supply two different limit distributions which are the shift of each
other with the above constant $C$. Indeed, if we consider the logarithms
of the characteristic functions of these limit distributions, then their
difference equals $it\int\frac{(\tau_a(u)-
\tau_{a'}(u))}{u^2}M(\,du)=itC$. On the other hand, if the
characteristic function of a random variable $\xi$ equals
$\varphi(t)$, then the characteristic function of the random variable
$\xi+C$  equals $e^{itC}\varphi(t)$.
 
Besides, we shall show that the limit of the measures
$M_k$ in Theorem~1 does not depend on the choice of the parameter
$a$. We have to settle the following problem. We have defined the
functions $M^\pm$, hence also the restriction of the measure $M$ 
to the half lines $(0,\infty)$ and $(-\infty,0)$ in formula~(1.3). 
We still have to define the measure $M(\{0\})$. This can be done 
with the help of formula~(1.6) and the identity 
$M(\{0\})=\limm_{s\to0} M([-s,s])$. The expression in 
formula~(1.6) depends on the constant $B_k=B_k(a)$ defined in 
formula~(1.5). We have to show that nevertheless the number 
$M(\{0\})$ does not depend on the parameter~$a$, and
$M(\{0\})\ge0$. Besides, we want to explain the probabilistic 
content of this quantity. This will be done in Remarks~2 and~3. 
Before these Remarks we show in Remark~1 that if formula (1.6) 
holds in such a point $s>0$ which is a point of continuity of 
the functions $M^+(\cdot)$ and $M^-(\cdot)$, then this relation 
also holds in all points of continuity $s'$ of these functions. 

\medskip\noindent
{\it Remark~1.}\/  If formula (1.6) holds in a point of 
continuity~$s$ of the $M^\pm(\cdot)$ functions, then 
relation~(1.3) implies that it holds in all points of continuity 
$s'>0$ of these functions. Indeed,
$$ \allowdisplaybreaks
\align
&M_k([-s',s'])-M_k([-s,s])=\int_{s}^{s'} u^2M^+_k(\,du)+
\int_{s}^{s'} u^2M^-_k(\,du)\\
&\qquad\to\int_{s}^{s'}M(\,du)+\int_{-s'}^{-s}M(\,du)
=M([-s',s'])-M([-s,s]).
\endalign
$$
This means that condition (1.6) can be replaced by its weakened version
which only states that it holds in one point of continuity of the
functions $M^\pm$.

\medskip \noindent
{\it Remark~2.}\/ Although the function $\tau(\cdot)=\tau_a(\cdot)$,
and as a consequence the constants $\beta_k=\beta_k(a)$, the norming
constants $b_k=b_k(a)$ and the constants $B_k=B_k(a)$ appearing in
formula~(1.6) depend on the parameter $a>0$, it can be shown that
$\limm_{k\to\infty}(B_k(a')-B_k(a))=0$ for two different constants
$a>0$ and $a'>0$. This means that formula (1.6) is meaningful, its
content does not depend on the choice of the parameter $>0$. To prove
this statement let us first observe that because of the condition of
uniform smallness $\limm_{k\to\infty}\supp_{1\le j\le n_k}|
\beta_{k,j}(a')-\beta_{k,j}(a)|=0$, and by relation (1.3)
$\supp_k\summ_{j=1}^{n_k} |\beta_{k,j}(a')-\beta_{k,j}(a)|<\infty$.
The last relation holds, since for all pairs $a'>a>0$
$$
\sum_{j=1}^{n_k}|\beta_{k,j}(a')-\beta_{k,j}(a)|\le
\int_a^{a'}u(M_k^+(\,du)+M_k^-(du))+(a'-a)[M^+_k(a')+M_k^-(a')],
$$
and the right-hand side of the last expression can be bounded because of
formula (1.3)  independently of~$k$.
 
Some difficulty arises in the proof, because the sequence
$B^*_k=\summ_{j=1}^{n_k}|\beta_{k,j}|$ may be unbounded. To overcome
this difficulty we can show that because of the uniform smallness
condition $\limm_{k\to\infty}\supp_{1\le j\le n_k}|\beta_{k,j}(a)|=0$.
Indeed, for all numbers $\e>0$ $|\beta_{k,j}(a)|\le
\e+aP(|\xi_{k,j}>\e)\le2\e$ if $k\ge k_0(\e,a)$ and $1\le j\le n_k$.
This relation holds, since $aP\(|\xi_{k,j}|>\e\)\le \e$ if $k\ge
k_0(\e,a)$. Hence $|B_k(a')-B_k(a)|\le I_k+2II_k$, where
$$
I_k=\summ_{j=1}^{n_k}|\beta_{k,j}(a')-\beta_{k,j}(a)|^2\le \sup_{1\le
j\le n_k} |\beta_{k,j}(a')-\beta_{k,j}(a)|\summ_{j=1}^{n_k}
|\beta_{k,j}(a')-\beta_{k,j}(a)|\to0,
$$
if $k\to\infty$, and
$$
II_k=\summ_{j=1}^{n_k}|(\beta_{k,j}(a')-\beta_{k,j}(a)) \beta_{k,j}(a)|
\le\supp_{1\le j\le n_k} |(\beta_{k,j}(a)|
\summ_{j=1}^{n_k}|\beta_{k,j}(a')-(\beta_{k,j}(a)|\to0
$$
if $k\to\infty$. This implies the statement of Remark~2.

\medskip\noindent
{\it Remark~3.}\/ Let us define, similarly to formula (1.4),
the function $\tau'(x)=\tau'_a(x)$ as $\tau'(x)=x$ if $|x|\le a$, and
$\tau'(x)=0$ if $|x|>a$. Set
$\beta'_{k,j}=\beta'_{k,j}(a)=E\tau'(\xi_{k,j})$,
$B'_k=B'_{k,j}(a)=\summ_{j=1}^{n_k}{\beta'_{k,j}}^2$. With a natural
modification of the argument in Remark~2 we get that
$\limm_{k\to\infty}(B_k-B_k')=0$, and here we can write $B'_k(a')$ with
an arbitrary number $a'>0$ instead of the number $B'_k(a)$. By
exploiting this fact and carrying out a limiting procedure $\e\to0$
through such numbers $\e$ which are points of continuity of both
functions $M^+$ and $M^-$ we get that
$$
\aligned
M(\{0\})&=\lim_{\e\to0}M([-\e,\e])=
\lim_{\e\to0}\lim_{k\to\infty}\(M_k([-\e,\e])-B'_k(\e) \) \\
&=\lim_{\e\to0}\lim_{k\to\infty}
\(\sum_{j=1}^{n_k}\(E\tau'_\e(\xi_{k,j})^2- (E\tau'_\e(\xi_{k,j}))^2\)\)
=\lim_{\e\to0}\lim_{k\to\infty}\sum_{j=1}^{n_k}\text{Var}\,
\tau'_\e(\xi_{k,j}).
\endaligned \tag1.8
$$
Formula (1.8) means in particular that $M(\{0\})\ge0$. The following
heuristic content can be given to formula~(1.8). The variance of the
normal component of the limit distribution, the number $M(\{0\})$ can be
obtained in the following way: We truncate the random variables in a
fixed row of the triangular array at a level~$\e>0$, we sum up these
random variables, and calculate the variance of the sum. Then take their
limit as the index of the row~$k$ tends to infinity and then the level
of the truncation $\e$ tends to zero. In an informal way this means
that the normal component of the limit distribution is the
``contribution of the inside part'' of the summands. The former argument
also shows that the expression defining the measure
$M([-s,s])$ in formula (1.6) is necessarily non-negative.
 
Formula (1.6) is equivalent to the relation $(1.6')$
$$
M(\{0\})=\lim_{\e\to0}\lim_{k\to\infty}\(M_k([-\e,\e])-B'_k(\e)\), \tag$1.6'$
$$
where such numbers $\e>0$ are taken in the limit which are points of
continuity of the measure~$M$. We have already seen that relation
(1.6) implies relation~$(1.6')$. To see the converse statement let us
first fix some small number $\e'>0$ write up the identity
$$
\align
&M_k([-s,s])-B_k-M([-s,s])=M_k([-s,s])-M_k([-\e',\e'])\\
&\qquad\qquad-(M([-s,s])-M([-\e',\e']))
+(B'_k(\e')-B_k)+(M_k([-\e',\e'])\\
&\qquad\qquad-B'_k(\e')-M(\{0\})-(M([-\e',\e'])-M\{0\}),
\endalign
$$
and estimate the right hand side of this identity if relation~($1.6'$)
holds. Observe that
$$
\limm_{k\to\infty}(M_k([-s,s])-M_k([-\e',\e'])-
(M([-s,s])-M([-\e',\e']))=0
$$
and $\limm_{k\to\infty}(B'_k(\e')-B_k)=0$. In the proof of these two
relations we only need formula (1.3) and do not apply formula~(1.6).
If we fix a number $\e>0$ and choose the number $\e'=\e'(\e)>0$
sufficiently small then we also can write
$M([-\e',\e'])-M\{0\}|\le \e$ and because of relation $(1.6')$
$$
\limsupp_{k\to\infty}|M_k([-\e',\e'])-M_k(\{0\})-B'_k(\e')|\le \e.
$$
These relations together imply that if relation $(1.6')$ holds, then
$$
\limsup_{k\to\infty}|M_k([-s,s])-B_k-M([-s,s])|\le 2\e.
$$
Since the last inequality holds for all $\e\to0$, relation $(1.6')$
implies relation (1.6).
 
Formula (1.6) can be better applied in the subsequent proofs, and
probably it is simpler to check it. On the other hand the heuristic
content of formula $(1.6')$ is more understandable.

\medskip
An important step in the proof of Theorem~1 is the investigation of the
question when a sequence of infinitely divisible distributions described
by means of the L\'evy--Hinchin formula converges in distribution and
what the limit distribution is. To formulate a result in this direction
let us first introduce the notion of (weak) convergence of canonical
measure. This notion is a natural modification of the convergence of
distribution functions.

\medskip\noindent
{\bf Definition of convergence of canonical measures.} {\it Let $M_n$,
$n=1,2,\dots$, and $M$ be canonical measures on the real line. We say
that the canonical measures $M_n$ converge (weakly) to the canonical
measure $M$ if
$$
\align
\lim_{n\to\infty}M_n^+(x)&=
\lim_{n\to\infty}\int_x^\infty\frac1{u^2}M_n(\,du)=
M^+(x)=\int_x^\infty\frac1{u^2}M(\,du), \\
\lim_{n\to\infty}M_n^-(x)&=
\lim_{n\to\infty}\int_{-\infty}^{-x}\frac1{u^2}M_n(\,du)
=M^-(x)=\int_{-\infty}^{-x}\frac1{u^2}M(\,du),
\endalign
$$
for all such numbers $x>0$ where the function $M^+(\cdot)$ or
$M^-(\cdot)$ is continuous, and
$$
\lim_{n\to\infty} M_n([a,b])=M([a,b])
$$
for all such numbers $-\infty<a<b<\infty$  where the measure $M$
is continuous. (This continuity of the measure means that
$M(\{a\})=M(\{b\})=0$.)} 

\medskip\noindent
{\it Remark~4.}\/ Similarly to the convergence of distribution functions
the convergence of canonical measures can be expressed by means of
convergence of integrals of an appropriate class of continuous
functions. A sequence of canonical measures $M_n$ converges (weakly) to
a canonical measure $M$ if and only if $\int f(u)M_n(du)\to \int
f(u)M(du)$ for all such continuous functions $f$ for which
$\supp_u\frac{|f(u)|}{1+u^2}<\infty$. Actually
this statement can be deduced from the analogous statement about
distribution functions if we observe that in the case when the limit
measure is not identically zero a sequence of canonical
measures $M_n$ converges weakly to a canonical measure $M$ if and only
if the corresponding probability measures $F_n=F_n(M_n)$ defined as
$F_n(\,dx)=\frac{M_n(\,dx)}{C_n(1+x^2)}$,
$C_n=\int\frac{M_n(\,dx)}{1+x^2}$ converge (weakly) to the
probability measure $F(\,dx)=\frac{M(\,dx)}{C(1+x^2)}$,
$C=\int\frac{M(\,dx)}{1+x^2}$. Since we do no need this result
we omit the details. 

\medskip
Now we formulate the result about the convergence of infinitely
distributions we shall need in the sequel.

\medskip\noindent
{\bf  Theorem~2.} {\it Let a sequence of infinitely divisible
distributions be given with characteristic functions
$\varphi_n(t)$, $t\in \bold R$. Let the (existing) logarithms 
of these characteristic functions be of the form
$$
\log \varphi_n(t)=\int_{-\infty}^\infty \frac{e^{itu}-1-it\tau(u)}{u^2}
M_n(\,du)+iB_nt, \quad n=1,2,\dots, \tag1.9
$$
where $M_n$, $n=1,2,\dots$, is a sequence of canonical measures, and the
function $\tau(u)=\tau_a(u)$ is defined in formula (1.4). These
infinitely divisible distributions converge in distribution if and only
if there exists a canonical measure $M$ such that the canonical
measures $M_n$ converge (weakly) to the canonical measure $M$, and also
the limit $B=\limm_{n\to\infty} B_n$ exists. If the limit exists,
then its characteristic function has a logarithm which can be given in
the form
$$
\log \varphi(t)=\int_{-\infty}^\infty \frac{e^{itu}-1-it\tau(u)}{u^2}
M(\,du)+iBt.
$$

\medskip}
By means of a simplified version of the proof of Theorem~2 the following
Theorem~$2'$ can also be proved. 

\medskip\noindent
{\bf Theorem~$2'$.} {\it Let the logarithm of the characteristic
function of an infinitely divisible distribution be of the form
$$
\log \varphi(t)=\int_{-\infty}^\infty \frac{e^{itu}-1-it\tau(u)}{u^2}
M(\,du)+iBt
$$
with a canonical measure $M$ and real number $B$. Then the
distribution function or what is equivalent to it the logarithm of
its characteristic function $\log \varphi(t)$ determines
the canonical measure $M$ and constant $B$.}

\medskip\noindent
{\it Remark.} In the formulation of Theorems~2 and~$2'$ we did not
exploit that the characteristic function of all infinitely 
divisible distributions can be given in the form of formula~(1.7). 
This result will be proved later with the help of an argument which
implicitly exploits these results.

\medskip
 Finally, we prove the following statement which is useful in the proof
of the  L\'evy--Hinchin formula. \medskip\noindent
{\bf Theorem~3.} {\it Let $\xi_{k,j}$, $k=1,2,\dots$, $1\le
j\le n_k$, be such a triangular array whose rows contain such random
variables which are not only independent but also identically
distributed. (But we do not assume that the triangular array satisfies
the uniform smallness condition.) Furthermore, let us assume that
$n_k\to\infty$ if $k\to\infty$, and the sums
$S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ converge in distribution. Then this
triangular array satisfies the uniform smallness condition.}

\medskip
The main result of this part of the work, Theorem~1, gave a necessary
and sufficient condition for the existence of a limit distribution for
the normalized sums of the elements $\xi_{k,j}$, $k=1,2,\dots$, $1\le
j\le n_k$, of the rows in a triangular array satisfying the
uniform smallness condition. With the help of a result contained in
Lemma~2 to be formulated later the  general problem considered in
Theorem~1 can be reduced to the special case when all elements of the
triangular array satisfy the condition $E\tau(\xi_{k,j})=0$,
$k=1,2,\dots$, $1\le j\le n_k$. Let us restrict our attention in the
following consideration to this special case. Then the relations
$E\beta_{k,j}=0$, $b_k=0$, $B_k=0$ hold for all indices $k=1,2,\dots$
and $1\le j\le n_k$ in formula (1.5) and the necessary and
sufficient condition for the existence of the limit distribution is
that the canonical measures $M_k$ defined in formula (1.1) converge
weakly to a canonical measure $M$. Besides, we can state that the
limit distribution is that infinitely divisible distribution whose
Poissonian part is determined by a Poisson process with counting
measure $u^{-2}M(du)$.
 
It is worthwhile to compare the result of Theorem~1 and Theorem~2.
Given a triangular array $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
$E\tau(\xi_{k,j})=0$ with distribution functions $F_{k,j}$
define a new triangular array whose elements are infinitely
divisible random variables $\eta_{k,j}$ and are determined as the
(regularized) sums of the elements of Poisson processes with counting
measures $F_{k,j}$. We also demand that the random variables
$\eta_{k,j}$ be independent for fixed $k$. The random variables
$\eta_{k,j}$ are called the associated random variables to the random
variables $\xi_{k,j}$. Let us compare the random sums
$S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ and $T_k=\summ_{j=1}^{n_k}\eta_{k,j}$.
Theorems~1 and~2 say that these random sums have a limit distribution at
the same time, and their limit agree. This agreement of the limit
distribution in these two cases is not a random coincidence. It has a
deeper reason.
 
In Part~III we shall give another proof of the sufficiency part of
Theorem~1. In that part of the work we shall prove the equiconvergence
in distribution of the above defined random sums $S_k$  and $T_k$ in a
direct probabilistic way by means of an appropriate coupling. Let us
give a short informal explanation for this equiconvergence. Let us split
both  the random variables $\xi_{k,j}$ and the random variables
$\eta_{k,j}$ to their inner and outer part as
$\xi_{k,j}=\xi_{k,j}I(|\xi_{k,j}|\le \e_k)
+\xi_{k,j}I(|\xi_{k,j}|> \e_k)$
and $\eta_{k,j}=\eta_{k,j}I(|\eta_{k,j}|\le \e_k)
+\eta_{k,j}I(|\eta_{k,j}|> \e_k)$ with some appropriate constants $\e_k$
which tend to zero sufficiently slowly. Then some calculation shows that
the sum of the inner parts of the random variables $\xi_{k,j}$ and
$\eta_{k,j}$ satisfy the central limit theorem with the same variance.
The sum of the outer part of the random variables $\xi_{k,j}$ and
$\eta_{k,j}$ behave similarly because of a different reason. In this
case the Poisson process which determines the random variable
$\eta_{k,j}$ contains no points whose absolute value is larger than
$\e_k$ with probability almost one because of the uniform smallness
condition. The probability of the event that for a fixed index $k$ one
of the Poisson processes determing the random variables $\eta_{k,j}$,
$1\le j\le n_k$, contains a point with absolute value larger than
$\e_k$ is not negligible, but the probability of the event that one of
these Poisson processes contains at least two such points is negligibly
small. This property enables us to couple the outer part of the random
variables $\xi_{k,j}$ and $\eta_{k,j}$ so that they are so close to
each other that even their sums are close.
 
The details of this rather sketchy argument will be worked out in
Part~III of this work. Such a study may help to understand better the
above results. Besides, the  coupling method worked out in Part~III
enables us to make a useful generalization. We shall prove with the help
of this method a functional limit theorem version of Theorem~1.
 
 
\beginsection 2. Some interesting consequences of the above results.
 
Most classical limit theorems of probability  which are related to the
behaviour of the distribution of  sums of independent random variables
can be deduced from the above results. We present some interesting
applications.
 
\medskip\noindent 
{\script A.) The L\'evy--Hinchin formula.}

\medskip\noindent
{\bf The L\'evy--Hinchin formula:} {\it
A  distribution function is infinitely divisible if and only if its
characteristic function $\varphi(t)$, $t\in\bold R^{(1)}$,  has a
logarithm which be written in the form
$$
\log \varphi(t)=\int_{-\infty}^\infty \frac{e^{itu}-1-it\tau(u)}{u^2}
M(\,du)+iBt \tag2.1
$$
where $M$  is a canonical measure on the real line, $B$ is a real
number and the function $\tau(\cdot)$ agrees with  the function defined
in formula (1.4) (with some fixed number $a>0$). In the representation
(2.1) of the characteristic function of an infinitely divisible
distribution the canonical measure $M$ and number $B$ is uniquely
determined.}

\medskip
{\it The proof of the L\'evy--Hinchin formula:}\/ In Part~I. of this
work we have already seen that formula (2.1) really defines the
logarithm of a characteristic function. Then it is not difficult to 
see that it is the logarithm of the characteristic function of an 
infinitely divisible distribution. Indeed, if the logarithm of the 
characteristic  function $\log \varphi(t)$ of a random variable 
$\xi$ is given by formula (2.1), then for arbitrary integer~$k$ 
its distribution equals the distribution of the sum of~$k$ 
independent and identically distributed random variables whose 
characteristic functions have a logarithm of the form 
$\frac{\log\varphi(t)}k$, i.e.\ it is given by formula (2.1) so
that the measure  $M(\cdot)$ is replaced by $\frac{M(\cdot)}k$ 
and the constant $B$ by $\frac Bk$.
 
Conversely, if $\xi$ is an infinitely distributed random variable, i.e.
for all integers~$k$ there exists $k$ independent and identically
distributed random variables $\xi_{k,1},\dots,\xi_{k,k}$ such that the
distribution of the sum $S_k=\xi_{k,1}+\cdots+\xi_{k,k}$ agrees with
the distribution of the random variable $\xi$, then by Theorem~3 the
triangular array $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le k$, satisfies
the uniform smallness condition. Then we can apply Theorem~1 which says
that there exists such a sequence of constant $b_k$ that the sequence
$S_k-b_k\overset\Delta\to=\xi-b_k$, where $\overset\Delta\to=$ denotes
equality in distribution, has a limit which can be given by formula
(1.7) by means of an appropriate canonical measure $M$. (We introduced
the norming constants $b_k$ in this argument, because with their help
we can achieve that the limit is given in formula~(1.7). The 
difference between formulas~(1.7) and~(2.1) is that the first 
formula does not contain an additive term $iBt$.) Since both
sequences $S_k$ and $S_k-b_k$ converge in distribution, the limit
$\limm_{k\to\infty}b_k=B$ exists, and formula (1.7) implies
relation (2.1).
 
\vfill\eject


\noindent
{\script B.) The central limit theorem.}

\medskip\noindent
We show that the results formulated in the first Section imply the most
important results about the central limit theorem. The most general, and
probably most known version of the central limit theorem states the
following result: Let $\xi_{k,j}$, $E\xi_{k,j}=0$,
$E\xi_{k,j}^2=\sigma^2_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
be a triangular array such that $\limm_{k\to\infty}
\summ_{j=1}^{n_k}\sigma_{k,j}^2=1$, and the following so-called
Lindeberg condition is satisfied: For all numbers $\e>0$
$\limm_{k\to\infty}\summ_{j=1}^{n_k}E\xi_{k,j}^2
I(|\xi_{k,j}|>\e)=0$, where $I(A)$ denotes the indicator function of the
set~$A$. Then the sums $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ converge in
distribution to the standard normal distribution as $k\to\infty$.
 
Also the following reversed statement formulated first by Feller
holds. Let $\xi_{k,j}$, $E\xi_{k,j}=0$, $E\xi_{k,j}^2=\sigma^2_{k,j}$,
$k=1,2,\dots$, $1\le j\le n_k$, be a triangular array such that
$\limm_{k\to\infty}\summ_{j=1}^{n_k}\sigma_{k,j}^2=1$. Let us also
demand the validity of the following condition which is slightly
stronger than the uniform smallness condition:
$\limm_{k\to\infty}\supp_{1\le j\le n_k}\sigma^2_{k,j}=0$.
If the random sums $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ converge
in distribution to the standard normal distribution, then this
triangular array also satisfies the Lindeberg condition. Moreover,
this statement also holds if instead of the convergence of the random
sums we only assume that the normalized sums $S_k-b_k$ converge in
distribution to the standard normal distribution with some appropriate
numbers $b_k$.
 
We prove the above results by means of the results formulated Section~1.
Actually, we shall prove a slightly stronger result. We prove that for
the validity of the Lindeberg condition for a triangular array which
satisfies the central limit theorem it is enough to assume the uniform
smallness condition instead of the relation
$\limm_{k\to\infty}\supp_{1\le j\le n_k}\sigma^2_{k,j}=0$.

First we show that the Lindeberg condition follows from the above
conditions. As we assumed the validity of the uniform smallness
condition we may apply Theorem~1. This result together with the
statement about the unique representation of infinitely divisible
distributions formulated in Theorem~$2'$ imply that if the central
limit theorem holds, then the canonical measures $M_k$ defined by
means of formula (1.1) from the distribution functions $F_{k,j}$ of the
random variables $\xi_{k,j}$ satisfy formulas (1.3) and (1.6) with the
constants $B_k$ defined in formula (1.5) and the canonical measure~$M$
such that $M(\{0\})=1$, $M(\bold R^1\setminus\{0\})=0$. As $B_k\ge0$,
hence by formula (1.6) for all numbers $\e>0$
$\limm_{k\to0}M_k([-\e,\e])-B_k=1$ with some number $B_k\ge0$. This
means that for all numbers $\e>0$
$\liminff_{j\to\infty}\summ_{j=1}^{n_k}
E\xi_{k,j}^2I(|\xi_{k,j}|\le\e)\ge1$. On the other hand,
$\limm_{j\to\infty}\summ_{j=1}^{n_k}E\xi_{k,j}^2=1$. Hence
$\limm_{j\to\infty}\summ_{j=1}^{n_k}E\xi_{k,j}^2 I(|\xi_{k,j}|>\e)=0$,
and this is the Lindeberg condition.
 
Conversely, we show that under the Lindeberg condition the central
limit theorem holds. In this case $\supp_{1\le j\le n_k}
P(|\xi_{k,j}|\ge\e)\le\frac1{\e^2}\summ_{j=1}^{n_k}
E\xi_{k,j}^2I(|\xi_{k,j}|>\e)\to0$ if $k\to\infty$. Hence the uniform
smallness condition holds, and we can apply Theorem~1. We have to show
that the  canonical measures $M_k$ constructed with the help of the
distribution functions $F_{k,j}$ of the random variables $\xi_{k,j}$,
$k=1,2,\dots$, $1\le j\le n_k$, satisfy the conditions (1.3) and (1.6)
with the measure $M$ defined by the relation $M\{0\})=1$, $M(\bold
R^1\setminus\{0\})=0$, and $\limm_{k\to\infty}b_k=0$. The Lindeberg
condition implies formula (1.3) with the above canonical measure $M$,
because for all numbers $a>0$ $M_k^+(a)=\summ_{j=1}^{n_k}
(1-F_{k,j}(a))\le \frac1{a^2}\summ_{j=1}^{n_k}E\xi_{k,j}^2
I(|\xi_{k,j}|>a)\to0$ if $k\to\infty$, and similarly
$\limm_{k\to\infty} M_k^-(a)=0$.
 
By the Lindeberg condition and the relation
$\limm_{k\to\infty}\summ_{j=1}^{n_k}E\xi_{k,j}^2=1$
$\limm_{k\to\infty}M_k([-s,s])=1$ for all $\e>0$. Hence to prove
formula (1.6) it is enough to show that $\limm_{k\to\infty}B_k=0$. This
relation holds since by the condition $E\xi_{k,j}=0$
$$
B_k=\summ_{j=1}^{n_k}\(E(\tau(\xi_{k,j})-\xi_{k,j})\)^2
\le\summ_{j=1}^{n_k}\(E|\xi_{k,j}|I(|\xi_{k,j}|>a)\)^2
\le\summ_{j=1}^{n_k}E\xi_{k,j}^2P^2(|\xi_{k,j}|>a),
$$
and this implies that $\limm_{k\to\infty}B_k=0$, since
$\limm_{k\to\infty}\summ_{j=1}^{n_k}E\xi^2_{k,j}=1$, and
$\limm_{k\to\infty}\supp_{1\le j\le n_k} P(|\xi_{k,j}>a)=0$.
Similarly,
$$
|b_k|=\left|\summ_{j=1}^{n_k}E\(\tau(\xi_{k,j})-\xi_{k,j}\)\right|
\le\summ_{j=1}^{n_k} E|\xi_{k,j}|I(|\xi_{k,j}|>a)
\le\frac1a\summ_{j=1}^{n_k}E\xi_{k,j}^2I(|\xi_{k,j}|>a),
$$
hence $\limm_{k\to\infty}b_k=0$ by the Lindeberg condition. This means
that by Theorem~1 the sums $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ converge in
distribution to a distribution function whose characteristic function
has the logarithm of the form $\log \varphi(t)=-\frac{t^2}2$. Hence
the central limit theorem holds under the above conditions.

\medskip\noindent
{\script C.) The weak law of large numbers.}

\medskip
Let $\xi_1,\xi_2,\dots$, be a sequence of independent, identically
distributed random variables, and consider the partial sums
$S_n=\summ_{k=1}^n\xi_k$, $n=1,2,\dots$,  made from these random
variables. A classical result of the probability deals with the problem
when these partial sums satisfy the weak law of large numbers, i.e.
under what conditions imposed for the distribution function
$F(x)$ of the random variables $\xi_k$ does  the relation
$\frac{S_n}n\Rightarrow a$ hold with some real number
$a$ as $n\to\infty$, where
$\Rightarrow$ denotes stochastic convergence. As stochastic convergence
of random variables to a number is equivalent to the convergence of
these random variables in distribution to the probability measure
concentrated in the point corresponding to this number the results
formulated in the first Section help us to answer this question. The
following result can be proved. 

\medskip\noindent
{\bf The weak law of large numbers.} {\it Let
$\xi_1,\xi_2,\dots$, be a sequence of independent and identically
distributed random variables with distribution function $F(x)$.
The averages $\frac
{S_n}n=\frac1n\summ_{k=1}^n\xi_k$ converge stochastically to a real
number~$a$ if the following two conditions are satisfied:

\medskip
\item {i.)} $\limm_{x\to\infty}  x[1-F(x)]=0$, and $\limm_{x\to\infty}
xF(-x)=0$
\item{ii.)} $\limm_{x\to\infty}\int_{-x}^x uF(\,du)=a$.

\medskip}\noindent
{\it Proof of the weak law of large numbers.}\/ Let us consider the
triangular array $\xi_{k,j}=\frac{\xi_j}k$, $k=1,2,\dots$,
$j=1,\dots,k$. It satisfies the condition of uniform smallness. The
weak law of large numbers means that the sums of the random
variables from a row of this triangular array converge to the
probability measure concentrated in the point~$a$. By Theorem~1 this
relation holds if and only if the measures $M_k(\,dx)=kx^2F(k\,dx)$,
the functions $M^+_k(x)=k(1-F(kx))$ an $M^-_k(x)=kF(-kx)$ together
with the numbers
$$
\align
b_k&=k\(\int_{-a}^a uF(k\,du)+a(1-F(ak)-F(-ak))\)\\
&=\int_{-ka}^{ka}u F(\,du)+ak\(1-F(ak)-F(-ak)\),
\endalign
$$
and $B_k=\frac{b_k^2}k$ satisfy the conditions (1.3) and (1.6) with a
limit measure $M$, $M(R^1)=0$, and $\limm_{k\to\infty}b_k=a$.
 
Some calculation shows that condition (1.3) in this case is equivalent
to Condition~i.). If Condition~i.) is satisfied, then Condition ii.) is
equivalent to  the relation $\limm_{k\to\infty}b_k=a$. Finally,  under
conditions i.) and ii.) also the relation (1.6) holds,  since in this
case $\limm_{k\to\infty} B_k=0$, and $\limm_{k\to\infty} M_k([-s,s])=0$.
Indeed, partial integration yields that
$M_k([-s,s])=\int_{-ks}^{ks} \frac{u^2}kF(\,du)
=s^2k\[(1-F(ks)+F(-ks)\]-\int_0^{ks}\frac{1-F(u)+F(-u)]}k\,du$, and this
implies that
$$
M_k([-s,s])\le s^2k\[(1-F(ks)+F(-ks)\]\to0  \quad\text{if}\quad
k\to\infty.
$$

\medskip\noindent
{\script D.) A limit theorem with Poissonian limit distribution.}

\medskip\noindent 
In Part~I we have formulated and in its Appendix we
have also proved a limit theorem where  the limit distribution was
Poissonian. Now  we show that this result is a  simple consequence of
Theorem~1 formulated in Section~1. The result is the following
statement.

\medskip\noindent
{\bf Limit theorem with Poissonian limit distribution.}
{\it Let $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$ be a triangular
array which satisfies the following conditions:
\item{1.)} The random variables $\xi_{k,j}$ take non-negative
integer values.
\item{2.)} $P(\xi_{k,j}=1)=\lambda_{k,j}$,
$\limm_{k\to\infty}\summ_{j=1}^{n_k}\lambda_{k,j}=\lambda>0$.
\item{3.)} $\supp_{1\le j\le n_k}\lambda_{k,j}\to0$ if $k\to\infty$,
and $\summ_{j=1}^{n_k}P\(\xi_{k,j}\ge2\)\to0$ if $k\to\infty$.
 
Then the random sums $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ converge in
distribution to a Poisson distribution with parameter $\lambda$
if $k\to\infty$.} 

\medskip\noindent 
{\it Proof of the limit theorem
with Poissonian distribution function.} The triangular array
$\xi_{k,j}$ satisfies the uniform smallness condition, hence Theorem~1
can be applied for instance with the choice $a=\frac12$ in the
definition of the function $\tau_a(\cdot)$. Then the conditions of
Theorem~1 hold with the limit canonical measure $M$ of the form
$M(\{1\})=\lambda$, $M(\bold R^1\setminus\{1\})=0$ and
$\limm_{k\to\infty}b_k=\frac12$. Hence the random sums $S_k-b_k$
converge to a limit distribution whose characteristic function has a
logarithm of the form $\log\varphi(t)=\lambda\(e^{it}-1-i\frac t2\)$.
This implies that the random sums $S_k$ converge to the Poisson
distribution with parameter $\lambda$. 

\medskip
Let us mention still another important and interesting application of
the results in Section~1. Let $\xi_1,\xi_2,\dots$ be a sequence of
independent and identically distributed random variables with some
distribution function $F(x)$, put $S_n=\summ_{k=1}^{n_k}\xi_k$,
$n=1,2,\dots$, and consider the normalized sums $\frac{S_n-B_n}{A_n}$
with some appropriate norming constants $A_n$ and $B_n$. The following
problems arise in a natural way. For which distribution functions $F$
can the norming constants $A_n$ and $B_n$ be chosen in such a way that
the above normalized sums converge in distribution? How should we
choose these norming constants? What kind of limit distributions can
appear? These questions can be answered completely, and the answers
lead to the notion of stable distributions. The solution of these
problems is also based on the results formulated in the first
Section. The reason we do not go into the details is that a
complete solution also requires some knowledge about the so-called
slowly varying functions, a subject we do not discuss here.

\medskip\noindent
{\bf 3. The proof of the results.}

\medskip\noindent
To prove the results formulated in Section~1 first we prove two
technical lemmas. In the first lemma we reformulate the condition of
uniform smallness in the language of characteristic functions. The
second lemma makes possible to reduce the proof of Theorem~1 to the
special case when the relation $E\tau(\xi_{k,j})=0$ holds for all
sufficiently large indices~$k$ and all numbers $1\le j\le n_k$. After
this we turn to the proof of the results. \medskip\noindent
{\script A.) The proof of two useful lemmas.}

\medskip\noindent
{\bf Lemma~1.} {\it Let $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
be a triangular array. Let $F_{k,j}$ denote the distribution and
$\varphi_{k,j}(t)=Ee^{it\xi_{k,j}}$ the characteristic function of the
random variable $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$. The above
triangular array satisfies the uniform smallness condition if and only
if for all numbers $K>0$
$$
\sup_{1\le j\le n_k}\sup_{|t|<K}\left|1-\varphi_{k,j}(t)\right|<\e
\quad \text{for } k\ge k_0(\e,K).
$$
}

\medskip \noindent
{\it The proof of Lemma~1.}\/ a.) Let us assume that the uniform
smallness condition is satisfied. Then with the choice
$\e'=\frac\e{2K}$
$$
\align
|1-\varphi_{k,j}(t)|&\le \int |1-e^{itx}|F_{k,j}(\,dx)=\int_{|x|\le
\e'}+\int _{|x|>\e'} \\
&\le \int_{-\e'}^{\e'}|tx|F_{k,j}(\,dx)+2P(|\xi_{k,j}|>\e')\le
K\e'+2P(|\xi_{k,j}|>\e')\le \e,
\endalign
$$
for $|t|\le K$ if $k\ge k_0(\e,K)$.

\medskip\noindent
b.) If the condition imposed for the characteristic functions
$\varphi_{k,j}$ holds then for all numbers $\e'>0$ and $K>0$
$$
\align
\e'&\ge\frac1{2K}\int_{-K}^K \Re\[1-\varphi_{k,j}(t)\]\,dt=
\frac1{2K}\int_{-K}^K \int_{-\infty}^\infty (1- \cos
tx)F_{k,j}(\,dx)\,dt \\
&=\frac1{2K} \int_{-\infty}^\infty 2K\[1- \frac{\sin
Kx}{Kx}\]F_{k,j}(\,dx)\ge \int_{\{|x|>\frac2K\}}\ge \frac12
P\(|\xi_{k,j}|>\frac2K\)
\endalign
$$
in the case $k\ge k_0(\e',K)$ and $1\le j\le n_k$. This relation 
with the choice $\e'=\frac\e2$ and $K=\frac2\e$ implies the 
uniform smallness condition of the triangular array. 

\medskip\noindent
{\bf Lemma~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
let us fix a number $a>0$ which appears in the definition of the
function~$\tau(\cdot)$ given in formula~(1.4). Then there exist such
numbers $\vartheta_{k,j}=\vartheta_{k,j}(a)$, $k=1,2,\dots$, $1\le j\le
n_k$, for which $\limm_{k\to\infty}\supp_{1\le j\le n_k}
|\vartheta_{k,j}|=0$, and the triangular array
$\xi'_{k,j}=\xi_{k,j}-\vartheta_{k,j}$ satisfies the condition
$E\tau(\xi'_{k,j})=E\tau_a(\xi'_{k,j})=0$ for all indices $k\ge
k_0=k_0(a)$, $1\le j\le n_k$, with an appropriate threshold index
$k_0(a)$, and the function $\tau(\cdot)=\tau_a(\cdot)$ defined in
formula~(1.4). Let $F'_{k,j}(x)=F_{k,j}(x+\vartheta_{k,j})$ denote the
distribution function of the random variable $\xi'_{k,j}$, and let us
define the canonical measures $M'_k$  and functions ${M'}^{\pm}_k(x)$
similarly to the measures $M_k$ and functions $M^\pm_k$ by means of
formulas (1.1) and (1.2) with the difference that we replace the
distribution functions $F_{k,j}$ with the distribution functions
$F'_{k,j}$ in these formulas.
 
The measures $M_k$ and functions $M^\pm_k$ satisfy the relations (1.3)
and (1.6) if and only if the measures $M'_k$ and ${M'}^\pm_k$ satisfy
them (with the same canonical measure $M$, but with the difference that
$B_k=0$ has to be written in formula (1.6) if the measure $M_k$ is
replaced by the measure $M'_k$.) If these relations hold, then the
numbers $b_k=\summ_{j=1}^{n_k}\beta_{k,j}$ and
$b'_k=\summ_{j=1}^{n_k}\vartheta_{k,j}$ satisfy the relation
$\limm_{k\to\infty}(b_k-b'_k)=0$.} 

\medskip
The last statement of Lemma~2 by which 
$\limm_{k\to\infty}(b_k-b'_k)=0$ implies that the random sums 
$S_k'=\summ_{j=1}^{n_k}\xi_{k,j}'$ defined with the help of the 
random variables $\xi_{k,j}'$ introduced in Lemma~2 and the 
normalized sums $S_k-b_k$ with $S_k=\summ_{j=1}^{n_k}\xi_{k,j}$ 
considered in Theorem~1 converge simultaneously in distribution. 
Moreover, the limit distributions of these expressions agree. Let 
us also remark that the necessary and sufficient condition of the 
convergence for the new triangular array $\xi_{k,j}'$ formulated 
in Theorem~1 means that the canonical measures $M'_k$ weakly 
converge to the canonical measure $M$. Let us observe that 
because of the relation 
$\limm_{k\to\infty}\supp_{1\le j\le n_k}|\vartheta_{k,j}|=0$ 
the uniform smallness condition for the triangular array 
$\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, implies that this 
condition also holds for the triangular array $\xi_{k,j}'$,
$k=1,2,\dots$, $1\le j\le n_k$. 

\medskip\noindent
{\it The proof of Lemma~2.}\/ Define the functions
$f_{k,j}(\vartheta)=E\tau(\xi_{k,j}-\vartheta)$. The uniform smallness
property of the triangular array implies that for an arbitrary small
number $\e>0$ $f_{k,j}(\e)<0$ and $f_{k,j}(-\e)>0$ if $k>k_0(\e)$ with
some threshold $k_0(\e)$. Indeed, we may assume that $a>2\e$. For all
sufficiently large indices $k$ the random variable $\tau(\xi_{k,j}-\e)$
is less than $-\e/2$, with probability almost one, and it is  less than
$a$ with probability one. This implies that $E\tau(\xi_{k,j}-\e)<0$ if
$k\ge k_0(\e)$. The inequality $E\tau(\xi_{k,j}+\e)>0$ can be proved
similarly. Because of these inequalities and the continuity of the
functions $f_{k,j}(\cdot)$ there exist numbers $\vartheta_{k,j}$ such
that $E\tau(\xi_{k,j}-\vartheta_{k,j})=0$ if $k\ge k_0$, and
$\limm_{k\to\infty}\supp_{1\le j\le n_k}|\vartheta_{k,j}|=0$.
 
As $|\vartheta_{k,j}|<\e$ holds for all sufficiently large indices~$k$
and all numbers $1\le j\le n_k$, the inequalities
$M_k^+(x-\e)<{M'}_k^+(x)<M_k^+(x+\e)$ and
$M_k^-(x-\e)<{M'}_k^-(x)<M_k^-(x+\e)$ hold if $k>k_0(\e)$. This implies
that in all points $x$ of continuity of the function $M^\pm(\cdot)$
the functions $M^\pm_k(x)$ and ${M'}^\pm_k(x)$ simultaneously converge
or do not converge to the function $M^\pm(x)$. This means that the
functions $M^\pm$ satisfy  the relation (1.3) if and only if the
function ${M'}^\pm$ satisfies it.
 
The identity 
$\tau(\xi'_{k,j})+\vartheta_{k,j}-\tau(\xi_{k,j})=
\xi'_{k,j}+\vartheta_{k,j}-\xi_{k,j}=0$ holds on the set 
$\{\oo\colon\;|\xi_{k,j}(\oo)|\le a-\e\}$ if $k\ge k_0(\e)$. Besides, 
the inequality
$|\tau(\xi'_{k,j})+\vartheta_{k,j}-\tau(\xi_{k,j})|\le 2\vartheta_{k,j}$
always holds, since $\tau(\cdot)$ is a Lipschitz~1 function. Applying
these results  and summing up for all indices $j$ we get that
$$
\align
\sum_{j=1}^{n_k}|\vartheta_{k,j}-\beta_{k,j}|&=
\sum_{j=1}^{n_k} |E\tau(\xi'_{k,j})+\vartheta_{k,j}-E\tau(\xi_{k,j})| \\
&=\sum_{j=1}^{n_k}
|E\(\tau(\xi'_{k,j})+\vartheta_{k,j}-\tau(\xi_{k,j})\)
I(|\xi_{k,j}|>a-\e)|\\
&\le2\sup_{1\le j\le n_k}|\vartheta_{k,j}|(M^+_k(a-\e)+M^-_k(a-\e)),
\endalign
$$
if $k\ge k_0(\e)$. Then we get, by taking the limit $k\to\infty$ we
get that $\limm_{k\to\infty}\summ_{j=1}^{n_k}
|\vartheta_{k,j}-\beta_{k,j}|=0$, and this is a stronger statement
then the relation $b_k-b'_k\to0$ as $k\to \infty$ formulated in
Lemma~2.
 
To complete the proof of Lemma~2 it is enough to show that if relation
(1.3) holds, then for all points of continuity $s>0$  of the functions
$M^\pm(\cdot)$
$$
\limm_{k\to\infty}(M_k([-s,s])-M'_k([-s,s])-B_k)=0
$$
Indeed, this means that the measures $M_k$ and $M'_k$  simultaneously
satisfy relation (1.6) in these points~$s$. (Observe that in formula
(1.6) the number $B'_k=0$ has to be taken from the measure $M'_k$.
This is so, because $B'_k=0$ for large $k$ since
$\beta'_{k,j}=E\tau(\xi'_{k,j})=0$ for all
indices $k\ge k_0$ and $1\le j\le n_k$.)
 
To prove this relation first we show that for all points of continuity
$s$ of the functions $M^\pm(\cdot)$
$$
\limm_{k\to\infty}\sum_{j=1}^{n_k}\(P(|\xi_{k,j}|>s)
-P(|\xi'_{k,j}|>s)\)=0, \tag3.1
$$
and
$$
\limm_{k\to\infty}\sum_{j=1}^{n_k}(E\tau(\xi_{k,j})^2
-E\tau(\xi'_{k,j})^2)-B_k=0.   \tag3.2
$$
 
As we consider a point of continuity $s$ of the functions $M^\pm$,
hence we get by applying formula (1.3) to the functions $M^\pm_k$ and
${M'}^\pm_k$ that
$$
\align
&\limm_{k\to\infty}\summ_{j=1}^{n_k}
(P(|\xi_{k,j}|>s)-P(|\xi'_{k,j}|>s))\\
&\qquad\qquad=\limm_{k\to\infty}
\((M_k^+(s)-{M'}^+_k(s))+(M_k^-(s)-{M'}_k^-(s))\)=0,
\endalign
$$
hence relation (3.1) holds. On the other hand, for all
sufficiently large indices
$k$ and all numbers $1\le j\le n_k$
$\tau(\xi_{k,j})^2-\tau(\xi'_{k,j})^2+\vartheta^2_{k,j}
-2\vartheta_{k,j}\tau(\xi_{k,j})=
\xi_{k,j}^2-(\xi'_{k,j})^2+\vartheta^2_{k,j}
-2\vartheta_{k,j}\xi_{k,j}=0$ 
on the set
$\{\oo\colon\;|\xi_{k,j}(\oo)|<a-\e\}$, and this expression is always 
smaller than $\const |\vartheta_{k,j}|$. Hence by taking the absolute 
values of the appropriate expressions and summing them up in the 
variable $j$ we get that
$$
\align
&\left|\sum_{j=1}^{n_k} E\(\tau(\xi_{k,j})^2-E\tau(\xi'_{k,j})^2
+\vartheta^2_{k,j}-2\vartheta_{k,j}\beta_{k,j}\)\right|\\
&\qquad\le \const\(2M^+_k(a-\e)+2M^-_k(a-\e)\)\sup_{1\le j\le n_k}
|\vartheta_{k,j}|,
\endalign
$$
and exploiting this inequality we get that
$$
\align
&\left|\sum_{j=1}^{n_k}
E\(\tau(\xi_{k,j})^2-E\tau(\xi'_{k,j})^2)\)-B_k\right|=
\left|\sum_{j=1}^{n_k} E\(\tau(\xi_{k,j})^2-E\tau(\xi'_{k,j})^2
-\beta_{k,j}^2\)\right| \\
&\qquad\le \const\(2M^+_k(a-\e)+2M^-_k(a-\e)\)\sup_{1\le j\le n_k}
|\vartheta_{k,j}|+\sum_{j=1}^{n_k}(\vartheta_{k,j}-\beta_{k,j})^2.
\endalign
$$
As the sequences $M_k^\pm(a-\e)$, $k=1,2,\dots$, are bounded,
and as we have proved $\limm_{k\to\infty}\summ_{j=1}^{n_k}
|\vartheta_{k,j}-\beta_{k,j})|=0$, the right-hand side of the above
expression tends to zero as $k\to\infty$, hence relation (3.2) holds.
 
Let us denote by $\tau_s(\cdot)$ the version of the function
$\tau(\cdot)=\tau_a(\cdot)$ defined in formula~(1.4) if the
parameter~$a$ in formula~(1.4) is replaced by the number $s$. Then we
can write the identity
$$ \allowdisplaybreaks
\align
&M_k([-s,s])-M'_k([-s,s])-B_k=\summ_{j=1}^{n_k}\(E\tau_s(\xi_{k,j})^2
-E\tau_s(\xi'_{k,j})^2\)-B_k\\
&\hskip6.5truecm -s^2\summ_{j=1}^{n_k}(P(|\xi_{k,j}|>s)
-P(|\xi'_{k,j}|>s)) \\
&\qquad=\sum_{j=1}^{n_k} \(E\tau_s(\xi_{k,j})^2-E\tau(\xi_{k,j})^2\)-
\sum_{j=1}^{n_k}\(E\tau_s(\xi'_{k,j})^2-E\tau(\xi'_{k,j})^2\)\\
&\qquad\qquad +\sum_{j=1}^{n_k}
E\(\tau(\xi_{k,j})^2-E\tau(\xi'_{k,j})^2\)-B_k
-s^2\summ_{j=1}^{n_k}
(P(|\xi_{k,j}|>s)-P(|\xi'_{k,j}|>s)).
\endalign
$$
To complete the proof of Theorem~2 it is enough to show that the
expression at the right-hand side of this identity tends to zero as
$k\to\infty$. That part of this expression which is contained in the
second line tends to zero by formulas~(3.1) and (3.2). We still have
to understand the contribution of the terms obtained as the  function
$\tau_a$ was replaced by $\tau_s$. Then the desired statement follows
from the following estimations.
$$
\align
\summ_{j=1}^{n_k} \(E\tau_s(\xi_{k,j})^2-E\tau(\xi_{k,j})^2)\)
&=\int\frac{\tau_s(u)^2-\tau(u)^2}{u^2} M_k(du) \\
&=\int_{\min (a,s)}^\infty\(\tau_s(u)^2-\tau(u)^2\)
(M^+_k(\,du)+M^-_k(\,du))\\
&\qquad\to \int_{\min(a,s)}^\infty \(\tau_s(u)^2-\tau(u)^2\)
(M^+(\,du)+M^-(\,du)).
\endalign
$$
The expression $\summ_{j=1}^{n_k}
\(E\tau_s(\xi'_{k,j})^2-E\tau(\xi'_{k,j})^2)\)$
has the same limit. Thus we hove shown that
$M_k([-s,s])-M'_k([-s,s])-B_k\to0$ if $k\to\infty$ and completed the
proof of Lemma~2.

\medskip
The proof of the sufficiency part of Theorem~1, the proof of the
statement that the limit distribution exists if the conditions of
Theorem~1 hold is relatively simple, and it can be directly done. The
proof of the necessity part of Theorem~1 is harder and to carry it out
we shall need the result of Theorem~2. So we shall prove it. In the
proof of Theorem~2 we also prove a result formulated in Lemma~3. This
Lemma~3 will be useful also in later considerations. In the next part
we prove the above results together with Theorem~$2'$. 

\medskip\noindent
{\script B.) The proof of Theorems~2 and~$2'$ and of the sufficiency
part of Theorem~1.} 

\medskip
First we prove the sufficiency part of Theorem~1 i.e.\ the statement
that if conditions (1.3) and (1.6) hold then the normalized sums
$S_k-b_k$ converge in distribution to that distribution function whose
characteristic function has a logarithm of the form~(1.7).

\medskip\noindent
{\it The proof of the sufficiency part of Theorem~1.}\/ We have to show
that the logarithms of the characteristic functions of the normalized
random sums $S_k-b_k$ satisfies the relation
$$
\log E e^{it(S_k-b_k)}=\sum_{j=1}^{n_k}
\log\varphi_{k,j}(t)-it\beta_{k,j}\to
\log \varphi(t),\quad \text{if } k\to\infty
$$
for all $t\in \bold R^1$ where the function $\log \varphi(t)$ is
defined in formula (1.7). Let us observe that because of the uniform
smallness condition assumed for the triangular array $\xi_{k,j}$,
$k=1,2,\dots$, $1\le j\le n_k$, $|1-\varphi_{k,j}(t)|\le\e$ for all
$\e>0$ if $k\ge k_0(\e,t)$. Hence the logarithm of the function
$\varphi_{k,j}(t)$ is meaningful if $k\ge k(t)$.
 
Let us first restrict our attention to the case when
$\beta_{k,j}=E\tau(\xi_{k,j})=0$ for all sufficiently large $k$ and
$1\le j\le n_k$. We shall prove that in this case
$$
\align
\limsup_{k\to\infty}&\sum_{j=1}^{n_k}
\left|1-\varphi_{k,j}(t)\right|<\infty \\
\lim_{k\to\infty}&\sum_{j=1}^{n_k}\(1-\varphi_{k,j}(t)\)
=-\log\varphi(t).
\endalign
$$
This two relations imply the limit theorem in the present case, because
$|\log z+(1-z)|<2|z|^2<2\e |z|$ if $|1-z|<\e$ and
$\frac12>\e>0$, and the first relation together with the uniform
smallness condition imply that
$$
\lim_{k\to\infty}\sum_{j=1}^{n_k}
\left|\log \varphi_{k,j}(t)+(1-\varphi_{k,j}(t))\right|=0.
$$
We get with the help of the relation $E\tau(\xi_{k,j})=0$ that
$$
\align
\sum_{j=1}^{n_k}|1-\varphi_{k,j}(t)|&=
\sum_{j=1}^{n_k}|1-\varphi_{k,j}(t)+itE\tau(\xi_{k,j})|\le
\sum_{j=1}^{n_k}\int|1-e^{itx}+it\tau(x)|F_{k,j}(dx)\\
&\le\sum_{j=1}^{n_k}\(\int_{-a}^a\frac12 t^2x^2 F_{k,j}(dx)
+\int_{\{|x|>a\}}(2+a|t|)F_{k,j}(\,dx)\)\\
&=\frac12t^2M_k\{[-a,a]\}+(2+a|t|)(M_k^+(a)+M_k^-(a))<\const,
\endalign
$$
and this is the first statement we wanted to prove. The second statement
can be proved similarly, because
$$
\align
\sum_{j=1}^{n_k}(1-\varphi_{k,j}(t))&=
\sum_{j=1}^{n_k}(1-\varphi_{k,j}(t)+itE\tau(\xi_{k,j}))=
\sum_{j=1}^{n_k}\int(1-e^{itx}+it\tau(x))F_{k,j}(dx)\\
&= \int_{-\infty}^\infty
\frac{1-e^{itx}+it\tau(x)}{x^2}M_k(\,dx)=\int_{-K}^K+\int_{|x|>K}
\endalign
$$
for arbitrary number $K>0$. Let us fix some number
 $\e>0$.
If $K=K(\e)$ is sufficiently large, and the points $\pm K$ are points of
continuity of the measure $M$, then because of relation
(1.3) and the boundedness of the function
$1-e^{itx}+it\tau(x)$
$$
\left|\int_{\{|x|>K\}}\frac{1-e^{itx}+it\tau(x)}{x^2}M_k(\,dx)
\right|<\e
$$
if $k>k_0$, and $\left|\int_{\{|x|>K\}}\frac{1-e^{itx}
+it\tau(x)}{x^2}M(\,dx)\right|<\e$ if we replace the measures $M_k$ 
by the limit measure $M$. On the other hand, because of the 
convergence of the canonical  measures $M_k$ to the canonical 
measure $M$ and the continuity of the function
$\frac1{x^2}(1-e^{itx}+it\tau(x))$
$$
\int_{-K}^K\frac{1-e^{itx}+it\tau(x)}{x^2}M_k(\,dx)
\to \int_{-K}^K\frac{1-e^{itx}+it\tau(x)}{x^2}M(\,dx),
\quad \text{if } k\to\infty.
$$
The above results imply the sufficiency in the special case considered
above. The general case when $E\tau(\xi_{k,j})\neq0$ is also possible
can be deduced from the already proven case with the help of Lemma~2.
 
Indeed, we can apply the already proven part of Theorem~1 for the
triangular array $\xi'_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, defined
in Lemma~2. With the help of this lemma we get  that the random sums
$S_k-b'_k$ and $S_k-b_k$, where the numbers $b'_k$ were defined also in
Lemma~2 have a limit distribution if the condition of Theorem~1 holds.
Besides, the logarithm of the characteristic function of the limit
distribution is given by formula~(1.7).

\medskip\noindent
{\it The proof of Theorem~2.}\/ First we prove the simpler sufficiency
part, i.e.\ the statement that the weak convergence of the canonical
measures $M_n$ to the canonical measure $M$ together with the
convergence of the numbers $B_n\to B$ imply that the sequence of
infinitely divisible distribution defined in Theorem~2 with the help
of the canonical measures $M_n$ and constants $B_n$ converge to the
infinitely divisible distribution determined by the canonical measure
$M$ and constant~$B$. As the convergence of the characteristic
functions of distribution functions to the characteristic function of
a distribution function imply the convergence of these distribution
functions to the distribution function with the limit characteristic
function it is enough to show that for all real numbers~$t$
$$
\align
\log \varphi_n(t)&=\int_{-\infty}^\infty
\frac{e^{itu}-1-it\tau(u)}{u^2} M_n(\,du)+iB_nt\\
&\qquad\to \log \varphi(t)=\int_{-\infty}^\infty
\frac{e^{itu}-1-it\tau(u)}{u^2} M(\,du)+iBt,
\endalign
$$
if $n\to\infty$.
 
As $M_n\to M$, and the integrand in the integrals we have considered
satisfy the inequality
$\left|\frac{e^{itu}-1-it\tau(u)}{u^2}\right|<\frac{\const} {1+u^2}$
for all $\e>0$ there exist such numbers $K=K(\e)>0$ for which  $\pm K$
are points of continuity of the measure $M$,
$$
\left|\int_{\{|u|>K\}} \frac{e^{itu}-1-it\tau(u)}{u^2} M_n(\,du)
\right|\le\e \quad\text{if }n\ge n_0(\e),
$$
and a similar inequality holds if we replace the measures $M_n$ by the
measure $M$. Besides, the convergence of the canonical measures
$M_n$ to the canonical measure $M$ and the continuity of the
integrand also implies that
$$
\int_{\{|u|<K\}} \frac{e^{itu}-1-it\tau(u)}{u^2} M_n(\,du)\to
\int_{\{|u|<K\}} \frac{e^{itu}-1-it\tau(u)}{u^2} M(\,du)
$$
if $n\to\infty$. These estimates and the relation $B_n\to B$ yield the
sufficiency part of Theorem~2 with a limiting procedure $\e\to0$.
 
To prove the necessity part of Theorem~1 let us first observe that if
(the (logarithms of) the characteristic functions of distribution
functions converge to a continuous function then the limit function is
(the logarithm of) the characteristic function of a distribution
function which is the limit of these distribution functions.
The main problem of the proof will be to decide when the functions
$\log\varphi_n$ defined in the formulation of Theorem~2 converge to a
continuous function and to describe the limit function.
 
To describe the limit of the sequence of functions
$\log\varphi_n(t)$ let us define the following ``smoothed version" of
these functions. Fix some number $h>0$ and put
$$
\psi_n(t)=\psi_n^h(t)=\log\varphi_n(t)-\frac1{2h}\int_{-h}^h
\log\varphi_n(t+s)\,ds.
$$
Then
$$
\aligned
\psi_n(t)&=\int _{-\infty}^\infty \(\frac{e^{itu}-1-it\tau(u)}{u^2}
-\frac1{2h}\int_{t-h}^{t+h}\frac{e^{isu}-1-is\tau(u)}{u^2}\,ds\)
M_n(\,du) \\
&=\int _{-\infty}^\infty\(\frac{e^{itu}}{u^2}-\frac1{2h}
\int_{t-h}^{t+h} \frac{e^{isu}}{u^2}\,ds\)
M_n(\,du)=\int_{-\infty}^\infty e^{itu}K(u)M_n(\,du),
\endaligned \tag3.3
$$
where
$$
K(u)=K_h(u)=\frac1{u^2}\(1-\frac{\sin hu}{hu}\). \tag$3.3'$
$$
We shall prove with the help of Lemma~3 formulated below that if the
infinitely divisible distribution functions considered in Theorem~2 have
a limit, then there exists a continuous function
$\bar\varphi(t)$ such that
$$
\limm_{n\to\infty}\psi_n(t)\to\psi(t)=\bar\varphi(t)-\frac1{2h}
\int_{-h}^h \bar\varphi(t+u)\,du. \tag3.4
$$
Indeed, by Lemma~3 if the functions $\log \varphi_n(t)$ are
characteristic functions of convergent distribution functions, then
they converge uniformly in all finite intervals to a continuous
function. Then we get formula (3.4) by taking a limit $n\to\infty$ in
the formula defining the function $\psi_n(\cdot)$ because of
Lebesgue's dominated convergence theorem and Lemma~3.
 
We have formulated Lemma~3 in such a way that we could also apply it
in the proof of Theorem~1. Before its formulation we make the following
remark.
 
Let $\log\varphi_n(t)$, $n=1,2,\dots$, be a sequence of functions
defined in formula~(1.9). By the results of Part~I we know that they are
the logarithms of the characteristic functions of infinitely divisible
distribution functions. If these distribution functions converge in
distribution then the functions $\log\varphi_n(t)$ converge uniformly
to a continuous function $\bar \varphi(t)$ in a small neighbourhood of
the origin. But we can state this uniform convergence --- at least
before a deeper investigation --- only in a small neighbourhood of the
origin. The above remark implies in particular that if the logarithms
of the characteristic functions defined in formula~(1.9) belong to
convergent distribution functions, then these distribution functions
satisfy the conditions of Lemma~3 formulated below.

\medskip\noindent
{\bf Lemma~3.} {\it Let $\log \varphi_n(t)$ be a sequence of functions
defined by formula~(1.9) with some canonical measures~$M_n$ and real
numbers~$B_n$. Let us assume that these functions
$\varphi_n(t)$ converge uniformly to a continuous function in a small
neighbourhood of the origin. (But we do not demand that the
distribution functions related to the functions
$\log \varphi_n(t)$ should converge in distribution.)
Then the functions $M_n$  taking part in the definition of the functions
$\log \varphi_n(t)$ satisfy the inequality
$$
\sup_{1\le n<\infty}\int_{-\infty}^\infty\frac1{1+u^2}M_n(\,du)<\infty.
\tag3.5
$$
In this case also the inequality
$$
\sup_{1\le n<\infty}\sup_{|t|<K}|\log\varphi_n(t)|<\infty  \tag3.6
$$
holds for all $K>0$.
 
If the functions $\log\varphi_n(t)$ are the logarithms of the
characteristic function of a\/ {\rm convergent} sequence of
distribution functions, then not only the characteristic functions of
these distribution functions but also their logarithms, the functions
$\log\varphi_n(t)$, converge uniformly to a continuous function in all
finite intervals.}  

\medskip\noindent
{\it The proof of Lemma~3.}\/ As the functions
$\log\varphi_n(t)$ are the logarithms of characteristic functions
of distribution functions, $\varphi_n(0)=1$, and under the conditions of
Lemma~3 there exists an appropriate $h>0$ such that the limit
$\limm_{n\to\infty}\log\varphi_n(t)=\bar\varphi(t)$ exists in the
interval $|t|\le h$, the limit is a continuous function, and the
convergence is uniform in this interval. For the sake of simpler
notations we fix such a number $h$ and we work with this number $h$
in the proof of Lemma~3 and Theorem~2 (thus for instance in the
definition of the already introduced function
$\psi_n(\cdot)=\psi_n^h(\cdot)$).
 
The uniform convergence of the functions $\log\varphi_n(t)$ in the
interval $[-h,h]$ together with their representation imply that there
exists an appropriate number $K>0$ such that
$$
\align
\infty&>K\ge2\sup_{n<\infty}\sup_{|t|\le h}|\log\varphi_n(t)|\ge
\sup_{n<\infty}\left| \log\varphi_n(0)-
\frac1{2h}\int_{-h}^h\log\varphi_n(t)\,dt\right|\\
&= \sup_{n<\infty} \left|\int_{-\infty}^\infty
\frac1{u^2}\(1-\frac{\sin hu}{hu}\)M_n(\,du)\right|
\ge C(h) \sup_{n<\infty} \left|\int_{-\infty}^\infty
\frac1{1+u^2}M_n(\,du)\right|,
\endalign
$$
because
$$
\frac1{u^2}\(1-\frac{\sin hu}{hu}\)=h^2\frac1{(hu)^2}\(1-\frac{\sin
hu}{hu}\)\ge \const\frac{h^2}{1+h^2u^2}\ge\const'\frac1{1+u^2}
$$
with some appropriate $\const$ and $\const'$ depending on $h$. Thus 
we have proved formula~(3.5). On the other hand, as
$$
\left|\frac{e^{itu}-1-i\tau(u)t}{u^2}\right|\le\cases
\frac{t^2}2, &\text{if }|u|<a\\
\frac{2+aK}{u^2}, &\text{if }|u|\ge a, \text { and } |t|\le K
\endcases
$$
formula (3.5) implies that the integral in the definition of the
function $\log\varphi_n(t)$ is uniformly bounded for a fixed finite
interval $|t|<K$ and all numbers $n=1,2,\dots$. The sequence
$B_n$ has to be finite, since otherwise the sequence
$\log\varphi_n(h)$ would be not bounded, thus it would not convergence.
The above argument implies relation (3.6). Finally the uniform
boundedness of the functions $\log\varphi_n(t)$ in finite intervals
imply that the functions $\varphi_n(t)$ and their limit, the function
$\varphi(t)$  is separated both from zero and infinity
in all finite intervals. Hence if the distributions determined
by the functions $\log\varphi_n(t)$ converge in distribution, then not
only the characteristic functions but the logarithms of the
characteristic functions of these distributions converge uniformly in
all finite intervals. Lemma~3 is proved.

\medskip
Let us turn back to the proof of Theorem~2. Let us introduce the
measures $\mu_n(\,du)=K(u)M_n(\,du)$, $n=1,2,\dots$, with the help of
the function $K(\cdot)\ge0$ defined in formula $(3.3')$. By rewriting
the  expression given for the function a $\psi_n(t)$ in formula
(3.3) by means of formula (3.4) we get that the Fourier transforms of
the measures $\mu_n$, the functions
$\psi_n(t)=\int_{\infty}^\infty e^{itu}\mu_n(\,du)$, converge to a
continuous function $\psi(t)$. Furthermore, the inequality
$$
\frac{C_1}{1+u^2}\le K(u)=\frac1{u^2}\(1-\frac{\sin hu}{hu}\)\le
\frac{C_2}{1+u^2}
$$
holds with some appropriate constants $C_2=C_2(h)>C_1=C_1(h)>0$. We
claim that the following two possibilities can appear. Either
$\limm_{n\to\infty}\psi_n(0)\allowmathbreak=0$, and in this case the
canonical measures $M_n$ weakly converge to the measure $M\equiv0$,
i.e.\ in this case $M(\bold R^1)=0$ or
$\limm_{n\to\infty}\psi_n(0)=\psi(0)>0$, and in this case the
probability measures $\bar\mu_n=\frac1{\psi_n(0)}\mu_n$ converge
weakly to a probability measure~$\mu$.
 
Indeed, if $\limm_{n\to\infty}\psi_n(0)=0$, then by the lower estimate
given for the function $K(u)$ we get that
$\limm_{n\to\infty}\int \frac1{1+u^2}M_n(\,du)=0$, i.e.\ the
measures $M_n$ converge weakly to the zero measure. If
$\lim\psi_n(0)>0$ (as $\psi_n(0)\ge0$, only the above two cases are
possible), then the above defined measures $\bar\mu_n$ are probability
measures and their characteristic functions converge to the continuous
function $\frac{\psi(t)}{\psi(0)}$. Hence in this case the measures
$\bar\mu_n$ converge weakly to a probability measure $\mu$. Let us
finally remark that because of the lower bound given for the function
$K(u)$ the continuity of the function $\frac{K(u)}{1+u^2}$ and the
relation $\limm_{n\to\infty}\bar\mu_n=\mu$ the canonical measure
$M_n(du)=K^{-1}(u)\mu_n(du)$ converge weakly to the canonical measure
$M(du)=K^{-1}(u)\psi(0)\mu(du)$.
 
Let us finally remark that, as we have already seen in the first part
of the proof, the convergence of the canonical measures $M_n$ to the
canonical measure $M$ implies that the integral parts of the formulas
expressing the functions $\log\varphi_n(t)$ converge to the integral
part of the formula expressing the function $\log\varphi(t)$. As the
weak convergence of the distributions considered in Theorem~2 implies
that the functions $\log\varphi_n(t)$ converge to the function
$\log\varphi(t)$, hence the constants $B_n$ in these formulas should
converge to the constant~$B$. Theorem~2 is proved. 

\medskip
Theorem~$2'$ can be proved similarly to the necessity part of Theorem~2.

\medskip\noindent
{\it The proof of Theorem~$2'$.}\/ Let us define, similarly to the
argument in Theorem~2, the function
$$
\psi(t)=\psi^h(t)=\log\varphi(t)-\frac1{2h}\int_{-h}^h
\log\varphi(t+u)\,du.
$$
Then
$$
\psi(t)=\int _{-\infty}^\infty e^{itu}K(u)M(du)
$$
with $K(u)=\frac1{u^2}\(1-\frac{\sin hu}{hu}\)$. The function
$\varphi(t)$ determines also the function $\psi(t)$. If $\psi(0)=0$,
then the measure $M$ is identically zero. If $\psi(0)>0$, then
$\bar\mu(\,du)=\frac{K(u)M(\,du)}{\psi(0)}$ is the uniquely determined
probability measure whose characteristic function is
$\frac{\psi(t)}{\psi(0)}$. Then the formula
$M(\,du)=\frac{\psi(0)}{K(u)}\bar\mu(\,du)$ also determines the
measure $M$. Finally, the function $\varphi(t)$ and the measure
$M$ also determine the constant $B$ in the formula expressing the
function $\varphi(t)$.
 
\medskip\noindent
{\script C.) The proof of the necessity part of Theorem~1.}

\medskip
Let us first briefly explain the idea of the proof. Lemma~2 enables us
to reduce the problem to the case when the summands $\xi_{k,j}$, $k\ge
k_0$, $1\le j\le n_k$, satisfy the identity $E\tau(\xi_{k,j})=0$ with
some threshold index~$k_0$. If the sums of the random variables
$\xi_{k,j}$ converge in distribution, then their characteristic
functions, the products $\prodd_{j=1}^{n_k}\varphi_{k,j}(t)$, converge
to a characteristic function $\psi(t)$. It is natural to take
logarithm in this relation. Then, since $\varphi_{k,j}(t)\sim1$
because of the uniform smallness condition one expects that the
approximation $\log\varphi_{k,j}(t)\sim\varphi_{k,j}(t)-1$ causes a
small error. These considerations suggest that $\limm_{k\to\infty}
\summ_{j=1}^{n_k}(\varphi_{k,j}(t)-1-itE\tau(\xi_{k,j}))
=\limm_{k\to\infty}\summ_{j=1}^{n_k}(\varphi_{k,j}(t)-1)=\log\psi(t)$.
Then if we write $\varphi_{k,j}(t)-1-itE\tau(\xi_{k,j})= \int\frac
{e^{itx}-1-it\tau(x)}{x^2}x^2 F_{k,j}(\,dx)$,  sum up these
identities for the argument $j$, then the last relation together with
the necessity part of Theorem~2 enable us to prove the desired result.
 
Nevertheless, all steps of the above argument demand a more detailed
justification. This is done in the proof below where we first consider
a simpler special case. Then we reduce the general case to it by means
of a technique called the symmetrization in the literature.

\medskip\noindent
{\it Proof of the necessity part of Theorem~1.}\/ Let us first consider
the special case when all random variables have symmetric distribution,
i.e.\ when the distribution functions of the random variables
$\xi_{k,j}$ and $-\xi_{k,j}$ agree, and the sequence of random sums
$S_k$, $k=1,2,\dots$, has a limit distribution.
 
If the sequence of random sums $S_k$ converges in distribution, then
the characteristic functions $\varphi_{k,j}(t)$ of the random variables
$\xi_{k,j}$ satisfy the relation
$$
\limm_{k\to\infty}\psi_k(t)= \limm_{k\to\infty}
\prodd_{j=1}^{n_k}\varphi_{k,j}(t)=\psi(t) \tag3.7
$$
with a continuous function $\psi(t)$ which is the characteristic
function of the limit distribution. This relation implies that
$$
\lim_{k\to\infty}\log\psi_k(t)=\lim_{k\to\infty}\sum_{j=1}^{n_k}
\log\varphi_{k,j}(t)=\log\psi(t). \tag3.8
$$
in an appropriate interval $|t|\le h$. For the time being we cannot
prove this statement for all $t\in\bold R^1$, since we do not know that
the function $\psi(\cdot)$ in no points takes the value zero.
 
Because of the symmetry of the distribution functions of the random
variables $\xi_{k,j}$ $\varphi_{k,j}(t)=E \cos (t\xi_{k,j})$ is a real
number, and $-1\le\varphi_{k,j}(t)\le1$ for all numbers $t\in \bold
R^1$. Hence $1-\varphi_{k,j}(t)=|1-\varphi_{k,j}(t)|$. By Lemma~1 the
uniform smallness condition imposed on the triangular array
 $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, implies that for all
numbers $K>0$ we have $\limm_{k\to\infty}\supp_{|t|<K}\supp_{1\le j\le
n_k} |1-\varphi_{k,j}(t)|=0$. Hence for all numbers $\e>0$
$|\log\varphi_{k,j}(t)+(1-\varphi_{k,j}(t))|\le\e|1-\varphi_{k,j}(t)|$
if $k\ge k_0(\e)$, and we can write
$$
\lim_{k\to\infty}\log\bar\psi_k(t)=\lim_{k\to\infty}
\sum_{j=1}^{n_k}(\varphi_{k,j}(t)-1)=\log\psi(t), \tag$3.8'$
$$
instead of the relation (3.8) with the function
$\bar\psi_k(t)=\prodd_{j=1}^{n_k}e^{\varphi_{k,j}(t)-1}$
if $|t|\le h$. Furthermore,
$$
\aligned
\log\bar\psi_k(t)=\sum_{j=1}^{n_k}(\varphi_{k,j}(t)-1)
&=\sum_{j=1}^{n_k}\int (e^{itx}-1-it\tau(x))F_{k,j}(\,dx)\\
&=\int \frac{e^{itx}-1-it\tau(x)}{x^2}M_k(\,dx),
\endaligned \tag3.9
$$
because of the symmetric distribution of the random variables
$\xi_{k,j}$, where the function $\tau(x)$ and the measure $M_k$ are the
quantities defined in Theorem~1. This relation and formula $(3.8')$
imply that Lemma~3 can be applied for the functions $\bar\psi_k(t)$.
Hence those versions of formulas (3.5) and (3.6) hold where the
measures $M_n$ are replaced by the measures $M_k$ and
$\log\bar\psi_k(t)$ is written instead of $\log\varphi_n(t)$. This
version of formula~(3.6) implies that
$$
\supp_{1\le k<\infty}\supp_{|t|\le K}\summ_{j=1}^{n_k}
(1-\varphi_{k,j}(t))=\supp_{1\le k<\infty}\supp_{|t|\le K}
\summ_{j=1}^{n_k}|1-\varphi_{k,j}(t)|<\infty,
$$
for all numbers $K>0$, and $0\le\supp_{1\le k<\infty}\supp_{|t|\le
K}-\summ_{j=1}^{n_k}\log\varphi_{k,j}(t)<\infty$ because
$|\log\varphi_{k,j}(t)+(1-\varphi_{k,j}(t))|<(1-\varphi_{k,j}(t))$
if $|t|\le K$ and $k\ge k_0(K)$. Hence we can take logarithm in formula
(3.7) for all numbers $t\in \bold R^1$, and relations (3.8) and
$(3.8')$ hold for all $t\in \bold F^1$. This means that the functions
$\bar\psi_k(t)$ are the characteristic functions of such
(infinitely divisible) distributions which converge in distribution.
Hence Theorem~2 can be applied for these functions, and it yields
together with formula~(3.9) that relations (1.3) and (1.6) hold
(the latter one with $B_k=0$ for all indices~$k$) in the above
considered case.
 
In the next step we prove the necessity part of Theorem~1 in the case
when $E\tau(\xi_{k,j})=0$ for all $k\ge k_0$ and $1\le j\le n_k$ with
an appropriate threshold index~$k_0$, and the normalized sums of the
random variables from fixed rows of the triangular array we consider,
the random variables $S_k-\bar b_k$, converge in distribution with an
appropriate norming sequence~$\bar b_k$. We prove that in this case
the canonical measures $M_k$ constructed by means of the triangular
array $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, satisfy relations
(1.3) and (1.6).
 
In the proof of this statement we apply symmetrization of the random
variables we are working with, a technique useful in several
investigations of probability theory. That is, we consider a new
triangular array $\bar\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
independent of the original triangular array $\xi_{k,j}$,
$k=1,2,\dots$, $1\le j\le n_k$, and such that the random variables
$\xi_{k,j}$ and $\bar\xi_{k,j}$ have the same distribution. Then we
define the random sums $\bar S_k=\summ_{j=1}^{n_k} \bar\xi_{k,j}$
similarly to the random sums $S_k$ and consider the differences
$S_k-\bar S_k$. The convergence of the random sums $S_k-\bar b_k$ in
distribution implies the same convergence for the expressions $S_k-\bar
S_k$, and the latter random variables can be obtained as the sums of the
random variables of the triangular array $\eta_{k,j}=\xi_{k,j}
-\bar\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, in fixed rows.
Observe that the random variables $\eta_{k,j}$ are symmetrically
distributed, hence we have already proved the necessity part of
Theorem~1 for the triangular array consisting of these random
variables.
 
We shall prove with the help of the above symmetrization that
$$
\sup_{1\le k<\infty} M_k^\pm(s)<\infty\text{ for all numbers } s>0,
\qquad \lim_{K\to\infty}\sup_{1\le k<\infty} M_k^\pm(K)=0,
$$
and
$$
\sup_{1\le k<\infty}M_k([-a,a])<\infty.
$$
 
Indeed, let us define the measures $M^0_k$ and functions ${M^0}^\pm_k$
similarly to the measures $M_k$ and functions $M^\pm_k$ introduced to
the formulation of Theorem~1 with the difference that now we replace
the distribution functions $F_{k,j}$ of the random variables
$\xi_{k,j}$ by the distribution function $\bar F_{k,j}=
F_{k,j}*F^-_{k,j}$ of the random variables
$\eta_{k,j}=\xi_{k,j}-\bar\xi_{k,j}$ in the definition of these
quantities, where $*$ denotes convolution, and
$F^-_{k,j}(x)=1-F_{k,j}(-x)$ is the distribution function of the random
variable $-\xi_{k,j}$. Then relations (1.3) and (1.6) hold (with
constant $B_k=0$) with an appropriate canonical measure $M^0$  if we
replace the quantities $M^\pm_k$ and $M_k$ by the quantities
${M^0}^\pm_k$ and $M^0$. Besides, for all $\e>0$ there exists a
threshold index $k_0=k_0(\e)$ such that $1-\bar F_{k,j}(x-\e)
=P(\xi_{k,j}-\bar\xi_{k,j}>x-\e)>P(\xi_{k,j}>x)P(\bar\xi_{k,j}>-\e)
>(1-\e)(1-F_{k,j}(x))$ for $x>2\e$ and $k>k_0(\e)$ because of the
uniform smallness condition. A similar inequality holds for the
quantity $\bar F_{k,j}(-x)$. Summing up these estimates for all
$j=1,\dots,n_k$ we obtain that
$M^\pm_k(x)\le\frac1{1-\e}{M^0}^\pm_k(x-\e)$ if
$x\ge2\e$ and $k\ge k_0(\e)$. As $\supp_{k<\infty}{M^0}_k^\pm(x)
<\infty$ for all numbers $x>0$, and
$\limm_{K\to\infty}\supp_{k<\infty}{M^0}_k^\pm(K)=0$, the above
inequalities imply the validity of the relations formulated to the
functions $M^\pm_k$.
 
We prove an inequality useful for our purposes to estimate the quantity
$M_k([-a,a])$. In its proof we exploit that
$E\tau(\xi_{k,j})=E\tau(\bar\xi_{k,j})=0$, and the random variables
$\xi_{k,j}$,
and $\bar\xi_{k,j}$ are independent. Besides, the functions
$$
v(x)=v_a(x)=\cases a&\text{ha }x>a\\
0&\text{ha }-a\le x\le a \\
-a&\text{ha }x<-a
\endcases
$$
satisfy the following relations:
$\tau(x)-v(x)=x$ if $|x|\le a$, and
$\tau(x)-v(x)=0$ if $|x|>a$. Furthermore, $\tau(x)v(x)=v^2(x)$. Hence
$$ \allowdisplaybreaks
\align
\int_{-2a}^{2a}x^2 \bar F_{k,j}(dx)&=
\int\int_{\{(x,y)\colon\;|x+y|\le 2a\}} (x+y)^2F_{k,j}(\,dx)F^-_{k,j}(\,dy)\\
&\ge\int\int_{\{(x,y)\colon\;|x|\le a,\,|y|\le a\}}
(x+y)^2F_{k,j}(\,dx)F^-_{k,j}(\,dy) \\
&=\int\int_{\{(x,y)\colon\;|x|\le a,\,|y|\le a\}}
(x-y)^2F_{k,j}(\,dx)F_{k,j}(\,dy) \\
&=E\(\tau(\xi_{k,j})-v(\xi_{k,j})-(\tau(\bar
\xi_{k,j})-v(\bar\xi_{k,j}))\)^2\\
&=2E\tau(\xi_{k,j})^2+2Ev(\xi_{k,j})^2-2(Ev(\xi_{k,j}))^2
-4E\tau(\xi_{k,j})v(\xi_{k,j})\\
&\ge2\int_{-a}^{a}x^2 F_{k,j}(dx)-4a^2(1-F_{k,j}(a)+F_{k,j}(-a)),
\endalign
$$
since
$$
\(Ev(\xi_{k,j})\)^2=a^2(1-F_{k,j}(a)+F_{k,j}(-a))^2\le
a^2(1-F_{k,j}(a)+F_{k,j}(-a)),
$$
and
$$
Ev(\xi_{k,j})^2=E\tau(\xi_{k,j})v(\xi_{k,j})=a^2(1-F_{k,j}(a)+F_{k,j}(-a)).
$$
By summing up these inequalities for $j=1,\dots,n_k$ we get that
$$
M^0_k([-2a,2a])\ge 2M_k([a,a])-4a^2(M_k^+(a)+M_k^-(a)).
$$
We know that $\supp_{1\le k<\infty} M^0_k([-2a,2a])<\infty$, (the
measures $M^0_k$ satisfy relation (1.6) with the choice $B_k=0$), and
have also seen that $\supp_{1\le k<\infty} M^\pm_k(a)<\infty$, These
relations imply that the inequality $\supp_{1\le k<\infty}M_k([-a,a])
<\infty$ holds.
 
With the help of the estimates obtained for the quantities $M_k$ and
$M^\pm_k$ we prove that for all numbers $T>0$
$$
\sup_{1\le k<\infty}\sum_{j=1}^{n_k}|1-\varphi_{k,j}(t)|\le C(T)
\quad\text{if } |t|\le T \tag3.10
$$
with an appropriate constant $C(T)<\infty$. Indeed, for all numbers
$|t|\le T$
$$
\align
|1-\varphi_{k,j}(t)|&=\left|
\int_{-\infty}^\infty(1-e^{itx}+it\tau(x))F_{k,j}(\,dx)\right|
\le \int_{-a}^a|1-e^{itx}+itx|F_{k,j}(\,dx) \\
&\qquad\qquad\qquad\qquad\qquad +\int_{|x|>a}
|1-e^{itx}+ita|F_{k,j}(\,dx)\\
&\le\frac{t^2}2 \int_{-a}^a x^2 F_{k,j}(\,dx)
+(2+|t|a)\(F_{k,j}(-a)+[1-F_{k,j}(a)]\).
\endalign
$$
By summing up these formulas for all $1\le j\le n_k$ and by exploiting
the inequality $|t|\le T$ we get that
$$
\sum_{j=1}^{n_k}|1-\varphi_{k,j}(t)|\le\frac{T^2}2
M_k([-a,a])+(2+Ta)(M^+_k(a)+M^-_k(a)).
$$
This estimate together with the existence of a finite bound for
the numbers $M^\pm_k(a)$ and $M_k([-a,a])$ independent of the index~$k$
imply relation~(3.10).
 
The uniform smallness condition, imposed for the triangular array
$\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, and Lemma~1 imply that
$\limm_{k\to\infty}\supp_{|t|\le T}\supp_{1\le j\le n_k}
|1-\varphi_{k,j}(t)|=0$ for all numbers $t\in\bold R^1$. As a
consequence, the relation
$\limm_{x\to0}\frac{\log(1-x)+x}x=0$ with the choice
$x=1-\varphi_{k,j}(t)$ implies that
$$
\limm_{k\to\infty}\sup_{|t|\le T}\supp_{1\le j\le
n_k}\frac{\left|\log\varphi_{k,j}(t)
+\(1-\varphi_{k,j}(t)\)\right|}{\left|1-\varphi_{k,j}(t)\right|}=0
$$
By this formula and relation (3.10)
$$
\limm_{k\to\infty}\sup_{|t|\le T}
\summ_{j=1}^{n_k}\left|\log\varphi_{k,j}(t)
+\(1-\varphi_{k,j}(t)\)\right|\to0 \tag3.11
$$
for all numbers $T>0$.
 
The convergence of the sequence of random variables $S_k-\bar b_k$
in distribution implies that
$$
\lim_{k\to\infty}\prod_{j=1}^{n_k}\(e^{-it\bar b_k/n_k}
\varphi_{k,j}(t)\)=\psi(t), \tag3.12
$$
where $\psi(t)$ is the characteristic function of the limit
distribution. Furthermore, the convergence is uniform in all finite
intervals. We claim that we can take logarithm in the above relation,
that is
$$
\lim_{k\to\infty}\sum_{j=1}^{n_k}\(\log\varphi_{k,j}(t)
-\frac{it\bar b_k}{n_k}\)=\log \psi(t). \tag3.13
$$
 
To prove this formula let us observe that by relations (3.10)
and~(3.11)
$$
\supp_{t\colon\;|t|\le T}\supp_k\summ_{j=1}^{n_k}\log
|\varphi_{k,j}(t)|\le C(T) \quad\text{if } k\ge k_0 \tag3.14
$$
with some appropriate constant $C(T)<\infty$ and threshold index $k_0$.
 
Because of relation (3.14) for all numbers $T>0$ there exist such
constants $0<C_1<C_2<\infty$ and threshold index $k_0$ which satisfy the
relation $C_1\le\prodd_{j=1}^{n_k} |\varphi_{k,j}(t)|\le C_2$ for all
numbers $-T\le t\le T$. Hence relation (3.12) implies the following
weakened version of formula (3.13): For all numbers $\e>0$, $-T\le t\le
T$ and indices $k\ge k_0(\e,T)$ where $k_0=k_0(\e,T)$ is an appropriate
threshold index there exists an integer $m=m(k,t)$ such that
$\left|\summ_{j=1}^{n_k}\(\log\varphi_{k,j}(t)-\frac{itb_k}{n_k}\)
-\log\psi(t)-i2\pi m(k,t)\right|<\e$. (In this argument  we must be a
little careful, because the logarithm is not a one-valued function on
the complex plane. This is the reason for the appearance of the
integers $m(k,t)$ in the last relation.) But because both functions
$\log \psi(t)$ and $\summ_{j=1}^{n_k}\(\log\varphi_{k,j}(t)-
\frac{itb_k}{n_k}\)$ are continuous, $\log\psi(0)=0$,
$\summ_{j=1}^{n_k}\log\varphi_{k,j}(0)=0$, which implies that
$m(k,0)=0$, hence $m(k,t)=0$  for all numbers $-T\le t\le T$ and
indices $k\ge k_0$. This means that relation (3.12) implies formula
(3.13) also in its original form.
 
By relation (3.11) we can replace the functions $\log\varphi_{k,j}(t)$
by the functions $\varphi_{k,j}(t)-1$ in formula (3.13) In such a way
we get that
$$
\lim_{k\to\infty}\sum_{j=1}^{n_k}(\varphi_{k,j}(t)-1)-it\bar b_k
=\log\psi(t).
$$
This formula can be rewritten because of the identity
$E\tau(\xi_{k,j})=0$ as
$$
\lim_{k\to\infty}\int\frac{e^{itx}-1-it\tau(x)}{x^2}M_k(dx)-it\bar b_k
=\log\psi(t)
$$
with the canonical measure $M_k$ introduced in the formulation of
Theorem~1. Then Theorem~2 can be applied, and it yields that the
sequence of canonical measures $M_k$ (weakly) converges to a canonical
measure $M$. This means that relations~(1.3) and~(1.6) hold (with the
choice $B_k=0$), and this is the statement we wanted to prove. 
(Besides, the relation $\limm_{k\to\infty}b_k=b$ also holds, and this
implies that the normalization $\bar b_k=0$ is also applicable, i.e.\
the non-normalized random sums~$S_k$ also have a limit distribution.)
 
Thus we have proved the necessity part of Theorem~1 in the case when
$E\tau(\xi_{k,j})=0$ for all sufficiently large~$k$ and
$1\le j\le n_k$. The result in the general situation can be deduced
from this case by means of Lemma~2. Indeed, by Lemma~2 one can find a
number $\vartheta_{j,k}$ for all $k\ge k_0$ and $1\le j\le n_k$ with
an appropriate threshold index $k_0$ such that the random variables
$\xi'_{k,j}=\xi_{k,j}-\vartheta_{k,j}$ satisfy the identity
$E\tau(\xi'_{k,j})=0$, and $\limm_{k\to\infty}\supp_{1\le j\le
n_k}|\vartheta_{k,j}|=0$. Then the necessity part of Theorem~1 holds
for the triangular array $\xi'_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$.
(Put $\xi'_{k,j}=\xi_{k,j}$ for $k<k_0$ for the sake of definitiveness.)
Besides, Lemma~2 also states that this result also implies that
the original triangular array $\xi_{k,j}$, $k=1,2,\dots$, $1\le j\le
n_k$, satisfies relations (1.3) and (1.6), and the normalized random
sums $S_k-b_k$ have a limit distribution, i.e.\ we can apply the
norming constant in the way described in Theorem~1.

\medskip
In such a way we have proved Theorem~1. We finish Part~II of this work
with the proof of Theorem~3.

\medskip\noindent {\script D.) The proof of Theorem 3.}

\medskip\noindent
{\it The proof of Theorem~3.}\/ It seems to be more appropriate first
to translate this problem to the language of characteristic functions
and to study it that way. By Lemma~1 the result we want to
prove can be expressed in the language of characteristic functions in
the following way: If a sequence of characteristic functions
$\oo_k(t)$, $t\in \bold R^1$, $k=1,2,\dots$,  is given together
with a characteristic function $\varphi(t)$ and a sequence of positive
integers $n_k$, $\limm_{k\to\infty}n_k\to\infty$, in such a way
that $\limm_{k\to\infty}\oo_k^{n_k}(t)\to\varphi(t)$ for all real
numbers $t$, then $\limm_{k\to\infty}\supp_{|t|\le K}|1-\oo_k(t)|=0$
for all numbers $K>0$.
 
We shall prove this result with the help of the symmetrization
technique. Let us consider beside a sequence of independent,
identically distributed random variables $\xi_{k,j}$, $1\le j\le n_k$,
with characteristic function $\oo_k(t)$ a new  sequence of independent
random variables $\bar\xi_{k,j}$, $1\le j\le n_k$, which have the same
distribution as the random variables $\xi_{k,j}$, and let the sequences
of the random variables $\xi_{k,j}$ and $\bar\xi_{k,j}$ be independent.
Put $\eta_{k,j}=\xi_{k,j}-\bar\xi_{k,j}$. Then the random variables
$\eta_{k,j}$ have characteristic function $|\oo(t)|^2$, the sums
$\summ_{j^=1}^{n_k}\eta_{k,j}$  tend in distribution to a distribution
function with characteristic function $|\varphi(t)|^2$. Hence
$\limm_{k\to\infty}|\omega_k(t)|^{2n_k}\to|\varphi(t)|^2$.
 
First we prove the following auxiliary statement: For all finite
intervals $[-K,K]$ there exists a number $C=C(K)>0$ such that
$\limsupp_{k\to\infty}\supp_{|t|\le K}n_k(1-|\oo_k(t)|^2)\le C$ if
$|t|\le K$.
 
For sufficiently small $K>0$ this auxiliary statement holds. Indeed,
because of the continuity of the function $\varphi(t)$ and the relation
$\varphi(0)=1$ for all $\e>0$ there exists a number $K=K(\e)$ such that
$|1-\varphi(t)|\le\e$ if $|t|\le K$. Then the uniform convergence of the
characteristic functions implies that
$1\ge\liminff_{k\to\infty}\inff_{|t|\le K}|\oo_k(t)|^{2n_k}\ge1-2\e=
C_1>0$, hence $1-\frac{C_2}{n_k}\le|\oo_k(t)|^2\le 1$, and
$n_k(1-|\oo_k(t)|^2)\le C_3<\infty$ for all numbers $|t|\le K$ with
appropriate constants $C_1>0$, $C_2>0$ and $C_3>0$.
 
As the auxiliary statement we want to prove holds in a small
neighbourhood of the origin it is enough to show that if it holds in an
interval $[-K,K]$, then it also holds in the interval
a $[-2K,2K]$. We prove this with the help of the following estimation
where $G_k$ denotes that (symmetrical) distribution whose characteristic
function is $|\oo_k(t)|^2$. If $|t|\le K$, then
$$ \allowdisplaybreaks
\align
n_k(1-|\oo_k(2t)|^2)&=n_k\int (1-\cos
2tx)G_k(\,dx)=2n_k\int\(1-\cos^2 tx\)G_k(\,dx)\\
&\le4n_k\int (1-\cos tx)G_k(\,dx)=4n_k(1-|\oo_k(t)|^2)\le 4C,
\endalign
$$
and this implies that if the auxiliary statement holds in the interval
$[-K,K]$ with an upper bound $C$, then it also holds
statement in the interval $[-2K,2K])$ with a new constant $C'=4C$.
 
The auxiliary statement together with the condition
$\limm_{k\to\infty}n_k=\infty$ imply that
$\limm_{k\to\infty}(1-|\oo_k(t)|)=0$, and the convergence is uniform in
all finite intervals. Besides, it also implies that
$\liminff_{k\to\infty}\inff_{|t|\le K}|\oo_k(t)|^{n_k}>0$, and as a
consequence $\inff_{|t|\le K}|\varphi(t)|>0$ for all numbers $K>0$.
Indeed, the relation $0\le n_k(1-|\oo_k(t)|)<C<\infty$ holds, and it
implies that $|\oo_k(t)|^{n_k}>C'>0$ with an appropriate constant
$C'=C'(C)$ for all sufficiently large~$k$. Let us write the
characteristic functions $\oo_k(t)$ for $k\ge k_0$  with a sufficiently
large threshold index~$k_0=k_0(K)$ and the characteristic function
$\varphi(t)$ in the polar form $\oo_k(t)=|\oo_k(t)|e^{i u_k(t)}$ and
$\varphi(t)=|\varphi(t)|e^{iv(t)}$ in an interval $[-K,K]$. Such a
representation is possible, because if $k_0$ is sufficiently large then
all these characteristic functions are separated from zero in the
interval $[-K,K]$. We may also assume that $u_k(0)=v(0)=0$, and the
functions $u_k(t)$ and $v(t)$ are continuous in the interval $[-K,K]$.
(The last assumption means that we define the exponent in the polar
representation of the characteristic functions in the natural way. We
do not deteriorate the nice behaviour of the exponents by adding some
unnecessary number $i2\pi r$ with some integer $r$ to the functions
$u_k(\cdot)$ or $v(\cdot)$ in some points $t$.) We complete the proof
of Theorem~3 if we show that $\limm_{k\to\infty}\supp_{|t\le
K}|u_k(t)|=0$ for all numbers $K>0$. Indeed, this relation implies that
$\limm_{k\to\infty}\supp_{|t\le K}|\oo_k(t)-|\oo_k(t)||=0$ which
fact together with the auxiliary statement imply the relation
$\limm_{k\to\infty}\supp_{|t|\le K}|1-\oo_k(t)|=0$ we wanted to prove.
 
Let us fix an interval $[-K,K]$. We know that with above the notations
$u_k(0)=v(0)=0$, and the functions $u_k(t)$ and $v(t)$ are
continuous. Besides, the convergence of the characteristic
functions we consider and their separation from zero in finite
intervals imply that $\limm_{k\to\infty}n_ku_k(t)=v(t)$ in the interval
$[K,K]$, and the convergence is uniform since the convergence of
characteristic functions to a limit characteristic function is uniform
in all finite intervals. Hence the value of the function
$v(t)$ determines the value $n_kv_k(t)$ with a good accuracy for large
indices~$k$. Nevertheless, some problems arise at this step in the
proof, because the number $n_ku_k(t)$ determines the number $u_k(t)$
only modulo $\frac{2\pi}{n_k}$. It depends also on the behaviour of the
function $u_k(\cdot)$ in the interval $[0,t)$ how we have to define the
number $u_k(t)$.

This difficulty can be overcome if we exploit the uniform continuity
of the function $v(\cdot)$ and the uniform convergence of the functions
$n_ku_k(\cdot)$ to the function $v(\cdot)$ in the interval $[-K,K]$.
These properties imply that there exists some number
$\delta=\delta(K)>0$ such that
$\supp_{|t|\le K, |t-s|\le\delta}|v(t)-v(s)|\le\frac\pi3$, and
$\supp_{|t|\le K, |t-s|\le\delta}n_k|u_k(t)-u_k(s)|\le\frac\pi2$
for all sufficiently large indices~$k$. We claim that these facts
together with the continuity of the functions $u_k(t)$ imply that
$|u_k(t)-u_k(s)|\le \frac\pi{n_k}$  for all sufficiently large
indices~k if $|t-s|\le \delta$, $|s|\le K$ and $|t|\le K$.
 
Indeed, let us consider an interval $[s,t]\subset[-K,K]$ whose length
is not greater than~$\delta$. By the above facts the map $\bold
T_k(x)=n_ku_k(x)$ defined for $x\in[-K,K]$ maps such an interval $[s,t]$
to some interval $J$ shorter than $\pi$. Hence the set
$\{u_k(x)\colon\;x\in [s,t]\}$ is a subset of the union of the intervals
$\frac J{n_k}+l\frac{2\pi}{n_k}$, $l=1,2,\dots$, and these intervals
are disjoint because of their short lengths. Hence the continuity of the
function $u_k$ implies that the set  $\{u_k(x)\colon\;x\in [s,t]\}$ is
contained in one of the intervals $\frac J{n_k}+l\frac{2\pi}{n_k}$,
hence the inequality $|u_k(t)-u_k(s)|\le \frac\pi{n_k}$ holds under
the above conditions.
 
Hence $\supp_{|t|\le K}|u_k(t)|\le\frac{K\pi}{\delta n_k}$ for all
sufficiently large indices~$k$, and since $n_k\to\infty$ as $k\to\infty$
$\limm_{k\to\infty} \supp_{|t|\le K}|u_k(t)|=0$. Theorem~3 is proved.
 
\bye

 
