Limit theorems and infinitely divisible distributions. Part III Functional limit theorems. by Péter Major The Mathematical Institute of the Hungarian Academy of Sciences

Summary: In the third part of this work we prove the sufficiency part of the main result proved in the second part with a different method. We make a good coupling of the independent random variables for whose sums we want to prove a limit theorem with independent random variables with so-called associated distributions to the distributions of the original random variables. This coupling shows that the sums of the original random variables and the sums of the random variables with these associated distributions have the same limit behaviour. This proof helps us to understand better the picture behind the limit theorem we discuss. Beside this, its method enables us to prove a functional limit theorem version of this result.

1. Introduction. Formulation of the results.

In Theorem 1 of Part II of the work Limit theorems and infinitely divisible distributions we gave the necessary and sufficient condition for the convergence in distribution of the appropriate normalized sums of the random variables in a triangular array which satisfies the uniform smallness condition. We recall the sufficiency part of this result and give a new proof of this result which is based on a good coupling. Then, by applying this coupling argument again, we also prove a functional limit theorem version of this result. This functional limit theorem, whose exact formulation will be given later, states that under natural weak conditions not only the sums of the random variables in a row have a limit distribution, but also the distributions of the random broken lines, made from the partial sums in a natural way converge in distribution to a probability measure in the space of functions.

We will investigate the following result of Part II.

Theorem 1. Let ξ_{k,j}, be a triangular array with distribution functions F_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, which satisfies the uniform smallness condition. Let us introduce the canonical measures

M_{k}( dx) = ∑_{j=1}^{n_{k}} x^{2}F_{k,j}( dx), k = 1, 2, …,  (1.1)

fix some number a > 0, and define the function

τ(x) = τ_{a}(x) = \{ \begin{matrix}x & if |x| ≤ a \\ a & if x ≥ a \\ −a & if x ≤ −a \end{matrix} .  (1.2)

Péter Major

Let us assume that the random variables ξ_{k,j} satisfy the identity Eτ(ξ_{k,j}) = 0 for all indices k = 1, 2, …, 1 ≤ j ≤ n_{k}, and that the measures M_{k} converge to a canonical measure M_{0} on the real line.

Then the sums S_{k} = ∑_{j=1}^{n_{k}} ξ_{k,j} converge in distribution to a distribution function

whose characteristic function φ(t), or more explicitly its (existing) logarithm, is given by the formula

log φ(t) = ∫_{−∞}^{∞} \frac{e^{itu} − 1 − itτ(u)}{u^{2}}M_{0}( du),  (1.3)

where the canonical measure M_{0} is the limit of the canonical measures M_{k}, and the function τ was defined in formula (1.2).

To understand better the above result we recall some notions. A σ-finite measure M on the real line is called canonical if

M([−a, a]) < ∞, and ∫_{\{|x|>a\}} \frac{1}{x^{2}} M( dx) < ∞

for all real numbers a > 0. A sequence of canonical measures M_{k}, k = 1, 2, …, (weakly) converges to a canonical measure M_{0} if

lim_{k→∞} M_{k}^{+}(x) = lim_{k→∞} ∫_{x}^{∞} \frac{1}{u^{2}} M_{k}( du) = M_{0}^{+}(x) = ∫_{x}^{∞} \frac{1}{u^{2}} M_{0}( du),
lim_{k→∞} M_{k}^{−}(x) = lim_{k→∞} ∫_{−∞}^{−x} \frac{1}{u^{2}} M_{k}( du) = M_{0}^{−}(x) = ∫_{−∞}^{−x} \frac{1}{u^{2}} M_{0}( du),

(1.4)

for all such numbers x > 0 where the function M_{0}^{+}(·) or M_{0}^{−}(·) is continuous, and

lim_{k→∞} M_{k}{[a, b]} = M_{0}{[a, b]}

for all numbers − < ∞ < a < b < ∞ where the limit measure M_{0} is continuous. (The continuity of the measure M_{0} in the points a and b means that M_{0}({a}) = M_{0}({b}) = 0).

A triangular array ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, consists of random variables from which the random variables with fixed first index k are independent. It satisfies the uniform smallness condition if for all ε > 0 the relation lim_{k→∞} sup_{1≤j≤n_{k}} P (|ξ_{k,j}| > ε) = 0

holds.

To give a complete formulation of Theorem 1, or more explicitly of formula (1.3) we have to define the integrand in the integral of formula (1.3) also in the point u = 0. We do this by continuity arguments in the following way:

\frac{e^{itu} − 1 − itτ(u)}{u^{2}} |_{u=0} = lim_{u→0} \frac{e^{itu} − 1 − itτ(u)}{u^{2}} = −\frac{t^{2}}{2} .

Limit theorems and infinitely divisible distributions. Part III

The above formulated Theorem 1 contains only one part of Theorem 1 in Part II of this work, the sufficiency part which stated the existence of a limit distribution if the conditions of the theorem are satisfied. Moreover, even this result is formulated in the special case when the condition Eτ(ξ_{k,j}) = 0 holds for all indices k = 1, 2, … and 1 ≤ j ≤ n_{k}. But the general case can be reduced simply to this special case with the help of Lemma 2 in Part II. The formulation of the result is simpler in the special case considered here. Now the more complicated conditions about the behaviour of the measures M_{k} can be expressed as the weak convergence of the canonical measures M_{k} introduced in Theorem 1 to an appropriate canonical measure M_{0}.

In the problems of probability theory we often take from the terms of the random sum we investigate the expected value of these terms and in such a way we work with the sum of random variables with expectation zero. The condition Eτ(ξ_{k,j}) = 0 is a modified version of this property in the general case when the random variables we are working with may not have a finite expectation.

The proof of Theorem 1 applies a method essentially different from the method of Part II. We shall apply the following relatively simple result whose proof will be given at the end in the Appendix.

Theorem A. Let S_{k} and T_{k}, k = 1, 2, …, be sequences of such random variables for which the sequence of differences S_{k} − T_{k} stochastically converges to zero as k → ∞. If the sequence of random variables S_{k} converges in distribution to a distribution function F, then the sequence of random variables T_{k} converges in distribution to the same distribution function F.

Also the following generalization of this statement holds. Let a separable metric space (X, ρ) be given together with two sequences of random variables S_{k} and T_{k}, k = 1, 2, …, on a probability space which take their values on the space (X, ρ), and their distance ρ(S_{k}, T_{k}) tends to zero stochastically if k → ∞. It the sequence S_{k} converges weakly to a measure μ on the space (X, ρ), then the sequence of random variables T_{k} converges weakly to the same measure μ.

In the proof of Theorem 1 we only need the first statement of Theorem A. Its second statement is formulated because that will be needed in the proof of the functional limit theorem version of Theorem 1. The proof of Theorem 1 given here will be similar to the second proof of the Poisson limit theorem in the Appendix of Part I. We shall make a good coupling of the random variables considered in Theorem 1 with independent random variables with infinitely divisible distributions. We can make this coupling in such a way that the sums of the original and the sums of the coupled random variables are close to each other, and the sums of the counting measures of the Poisson measures whose (normalized) sums determine the infinitely divisible random variables we construct in this coupling construction is also convergent. Then Theorem A enables us to reduce the proof of Theorem 1 to the proof of the convergence of the coupled random variables in distribution. This latter statement can also be proved by means of a good coupling procedure and Theorem A.

To carry out the program sketched above it is useful to decompose the measures M_{k} appearing in the formulation of Theorem 1 to two terms in such a way that the first term is responsible for the convergence of the Gaussian and the second term for the convergence of the Poissonian part in the limit theorem we consider. The subsequent Lemma 1 supplies such a decomposition.

Lemma 1. Let ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, be a triangular array satisfying the uniform smallness condition. Let F_{k,j} denote the distribution function of the random variable ξ_{k,j}, and put G_{k,j}( dx) = x^{2}F_{k,j}( dx). Let us assume that the canonical mea-

sures M_{k} = ∑_{j=1}^{n_{k}} G_{k,j} converge weakly to a canonical measure M_{0} as k → ∞. Let us

write the limit measure M_{0} in the form M_{0} = M′_{0} + M′′_{0}, where M′_{0} is the restriction of the measure M_{0} to the origin, i.e. for all measurable sets A ⊂ R^{1} M′_{0}(A) = 0, if 0 ∉ A, and M′_{0}(A) = M_{0}({0}) if 0 ∈ A. Furthermore M′′_{0} = M_{0} − M′_{0}. Then there exists such a sequence of numbers ε_{k} > 0, ε_{k} → 0 if k → ∞, for which the measures M′_{k}, M′_{k}(A) = M_{k}(A ∩ I({|x| < ε_{k}}) converge weakly to the canonical measure M′_{0}, and the canonical measures M′′_{k}, M′′_{k}(A) = M_{k}(A ∩ I{|x| ≥ ε_{k}}) converge weakly to the canonical measure M′′_{0}, where I(B) denotes the indicator function of the set B. Furthermore, lim_{k→∞} sup_{1≤j≤n_{k}}(1 − F_{k,j}(ε_{k})) = 0, and lim_{k→∞} sup_{1≤j≤n_{k}} F_{k,j}(−ε_{k}) = 0, and even the relation

lim_{k→∞} ∑_{j=1}^{n_{k}}[(1 − F_{k,j}(ε_{k})) + F_{k,j}(−ε_{k})]^{2} = 0  (1.5)

holds.

We describe the coupling construction which enables the reduction of Theorem 1 to the proof of the convergence of appropriate infinitely divisible distributions. Then we shall also give the ideas behind this construction and formulate Proposition 1 which tells the most important properties of this coupling construction. Because of some reason of convenience we shall work in the coupling construction not with the random variables ξ_{k,j} considered in Theorem 1, but we shall construct instead new (for fixed index k independent) random variables with the same distributions.

Let the random variables ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, of the triangular array we consider satisfy the conditions of Theorem 1. For all pairs of indices k = 1, 2, …, 1 ≤ j ≤ n_{k}, let us consider the random variables ξ_{k,j} and define the probability measures ν̄_{k,j} and ¯ ν̄_{k,j} given by the formulas ν̄_{k,j}(A) = P (ξ_{k,j} ∈ A| |ξ_{k,j}| < ε_{k}) and ¯ ν̄_{k,j}(A) = P (ξ_{k,j} ∈ A| |ξ_{k,j}| ≥ ε_{k}) together with the numbers p_{k,j} = P (|ξ_{k,j}| ≥ ε_{k}) where A ∈ R^{1} are arbitrary measurable sets and the numbers ε_{k} are chosen in such a way that the results of Lemma 1 hold with them. Let η′_{k,j}, j = 1, …, n_{k}, be ν̄_{k,j} distributed random variables which are independent for fixed index k. Let us also consider a sequence of Poisson distributed random variables ζ_{k,j} with parameters p̄_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, where the number p̄_{k,j} is the solution of the equation 1 − e^{−p̄_{k,j}} = p_{k,j}. Let us also assume that these random variables ζ_{k,j} are independent for a fixed index k, and they are also independent of the random variables η′_{k,j}. Furthermore, let γ_{k,j,l}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, l = 1, 2, …, be random variables with distribution ¯ ν̄_{k,j} which are independent both from each other and the random variables defined before.

Let us define, beside the already constructed random variables η′_{k,j} the random

variables η′′_{k,j} = ∑_{l=1}^{ζ_{k,j}} γ_{k,j,l}, ξ′_{k,j} = η′_{k,j}I(ζ_{k,j} = 0), ξ′′_{k,j} = γ_{k,j,1}I(ζ_{k,j} ≥ 1), ξ̃_{k,j} = ξ′_{k,j} +

ξ′′_{k,j}, and η_{k,j} = η′_{k,j} + η′′_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}. We shall see that the random variables ξ̃_{k,j} and η_{k,j} constructed in such a way give a good coupling which satisfy Proposition 1 formulated below. This enables us to reduce the proof of Theorem 1 to the study of the sums of the random variables η_{k,j} which is a simpler problem.

The idea behind the above construction is the following. As we shall see the random variables ξ_{k,j} and ξ̃_{k,j} = ξ′_{k,j} + ξ′′_{k,j} have the same distributions. The random variables ξ′_{k,j} and η′_{k,j} are close to each other, hence they satisfy the central limit theorem with a Gaussian limit with the same expected value and variance. The reason we defined them in a slightly different way is that we wanted to achieve that the random variables η′_{k,j} and η′′_{k,j} be independent, because this allows to study their behaviour separately. The random variables ξ′′_{k,j} and η′′_{k,j} are also sufficiently close to each other, but this closeness has a different reason.We can observe that both probabilities P (η′_{k,j} ≠ 0) and P (η′′_{k,j} ≠ 0) are small for large indices k, but we need more knowledge about their behaviour. Our construction guarantees that ξ′′_{k,j}(ω) = η′′_{k,j}(ω) on the set {ω: ζ_{k,j}(ω) ≤ 1}, and the set {ω: ζ_{k,j}(ω) ≥ 2} has very small probability for large indices k. This fact will guarantee that the above constructed coupling is good for our purposes.

Let us also observe that the sequence of random variables γ_{k,j,1}, …, γ_{k,j,ζ_{k,j}} (with a random number of elements) is a Poisson process with counting measure p̄_{k,j}¯ν̄_{k,j}. Hence,

as we have seen in Part I the random variable η′′_{k,j} = ∑_{l=1}^{ζ_{k,j}} γ_{k,j,l} is infinitely divisible, and

the logarithm of its characteristic function can be given by the formula

log φ_{k,j}(t) = log Ee^{itη′′_{k,j}} = ∫ (e^{itu} − 1)p_{k,j}¯ν̄_{k,j}( du) = \frac{p̄_{j,k}}{p_{j,k}} ∫_{\{|u|≥ε_{k}\}} \frac{e^{itu} − 1}{u^{2}}G_{k,j}( du),

(1.6)

where G_{k,j}( du) = u^{2}F_{k,j}( du) agrees with the measure G_{k,j} defined in Lemma 1.

We formulate the most important properties of the above constructed coupling in the following Proposition 1.

Proposition 1. Let the triangular array ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, satisfy the conditions of Theorem 1. Then the above coupling construction formulated after Lemma 1 has the following properties: The distribution of the random variables ξ̃_{k,j} = ξ′_{k,j} + ξ′′_{k,j} and ξ_{k,j} agree. The triangular arrays η′_{k,j} and η′′_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k} are independent, i.e. for a fixed index k the random vectors η′_{k,j}, 1 ≤ j ≤ n_{k}, and η′′_{k,j}, 1 ≤ j ≤ n_{k}, are independent. Also the identities P (|ξ′_{k,j}| ≤ ε_{k}) = P (|η′_{k,j}| ≤ ε_{k}) = 1 hold with the same sequence of numbers ε_{k}, k = 1, 2, …, which appears in Lemma 1. Furthermore,

lim_{k→∞} ∑_{j=1}^{n_{k}} |Eξ′_{k,j} − Eη′_{k,j}| = 0,  (1.7)

and also the relations

\begin{matrix}sup_{1≤p≤n_{k}} |∑_{j=1}^{p}(ξ′_{k,j} − Eξ′_{k,j}) − (η′_{k,j} − Eη′_{k,j})| ⇒ 0, \\ sup_{1≤p≤n_{k}} |∑_{j=1}^{p}(ξ′′_{k,j} − Eτ(ξ′′_{k,j})) − (η′′_{k,j} − Eτ(η′′_{k,j}))| ⇒ 0 \end{matrix}  (1.8)

hold where ⇒ denotes stochastic convergence. The logarithm of the characteristic func-

tion of the sum ∑_{j=1}^{n_{k}}(η′′_{k,j} − Eτ(η′′_{k,j})) can be expressed by means of a canonical measure

M̄′′_{k} close to the canonical measure M′′_{k} as

log E exp \{it (∑_{j=1}^{n_{k}}(η′′_{k,j} − Eτ(η′′_{k,j})))\} = ∫ \frac{e^{itu} − 1 − itτ(u)}{u^{2}} M̄′′_{k}( du),  (1.9)

where M̄′′_{k}( du) = M′′_{k}( du) + ∑_{j=1}^{n_{k}} \frac{p̄_{k,j}−p_{k,j}}{p_{k,j}} G′_{k,j}( du), and G′_{k,j} is the restriction of the

measure G_{k,j} to the set R^{1} \ [−ε_{k}, ε_{k}].

The triangular array η′_{k,j} − Eη′_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, satisfies the central limit theorem with a Gaussian limit which has expectation zero and variance M_{0}({0}).

We formulate a result about the convergence of infinitely divisible distributions which enables us to complete the proof of Theorem 1. Let us recall the following result discussed in Part I of this work. If M is a canonical measure on the real line, ξ_{n}, n = 1, 2, …, is a Poisson process on the real line with counting measure μ(du) = \frac{M( du)}{u^{2}},

then the appropriate regularized version of the sum η = η_{M} = ∑_{n=1}^{∞} ξ_{n} − E ( ∑_{n=1}^{∞} τ(ξ_{n}))

of the points of the Poisson process, where τ(·) is the function introduced in formula (1.2) is convergent, and the random variable η_{M} has infinitely divisible distribution. More precisely, we can define the regularized sum by the formula

η_{M} = lim_{L→∞} (( ∑_{n: |ξ_{n}|>2^{−L}} ξ_{n}) − E ( ∑_{n: |ξ_{n}|>2^{−L}} τ(ξ_{n}))) .

and this limit is convergent with probability one. The distribution of the so defined random variable η_{M} has such a characteristic function whose logarithm equals

log φ(t) = log φ_{M}(t) = ∫ \frac{e^{itu} − 1 − itτ(u)}{u^{2}}M( du).  (1.10)

We shall call the random variable η_{M} defined in such a way the infinitely divisible random variable determined by the Poisson process ξ_{1}, ξ_{2}, … with counting measure μ. Now we formulate the following result.

Proposition 2. Let M_{k}, k = 1, 2, …, be a sequence of canonical measures which converges weakly to a canonical measure M_{0}. Let us also assume that the relations M_{0}(\{0\}) = 0 and M_{k}(\{0\}) = 0, k = 1, 2, …, hold. Put μ_{k}(du) = \frac{M_{k}(du)}{u^{2}}, k = 0, 1, 2, . . .. Then we can define Poisson processes ξ_{k,1}, ξ_{k,2}, … with counting measure μ_{k} and Poisson processes ξ̄_{k,1}, ξ̄_{k,2}, … with counting measure μ_{0} in such a way that the random variables η_{k} with infinitely divisible distribution determined by the Poisson processes ξ_{k,1}, ξ_{k,2}, … (introduced e.g. before formula 1.10) and the random variables η̄_{k} with infinitely divisible distribution determined by the Poisson processes ξ̄_{k.1}, ξ̄_{k,2}, … satisfy the relation η_{k} −η̄_{k} ⇒ 0 where ⇒ denotes stochastic convergence. (Let us remark that the distributions of the random variables η̄_{k} do not depend on the index k.)

Remark: Proposition 2 and Theorem A together imply that if the measures M_{k}, k = 0, 1, 2, …, satisfy the conditions of Proposition 2, then the distributions of the random variables η_{k} defined in Proposition 2 converge in distribution to the distribution function whose characteristic function is given in formula (1.3). We also remark that with the help of some additional work a stronger version of Proposition 2 could also be proved. It is possible to make such a construction in which η̄_{k} = η̄, that is these random variables (and the Poisson processes determing them) do not depend on the index k. Further, it can be achieved that also the relation η_{k} − η̄_{k} → 0 hold with probability one. But for our purposes the weaker result formulated in Proposition 2 is as good as its above mentioned stronger version.

We show that Propositions 1 and 2 together with Theorem A imply Theorem 1. Let us consider the random variables η′_{k,j}, η′′_{k,j}, and η_{k,j} = η′_{k,j} + η′′_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, defined in the coupling construction given after Lemma 1 together with the sums

T_{k} = ∑_{j=1}^{n_{k}}(η_{k,j} − Eτ(η_{k,j})), T′_{k} = ∑_{j=1}^{n_{k}}(η′_{k,j} − Eη′_{k,j}) and T′′_{k} = ∑_{j=1}^{n_{k}}(η′′_{k,j} − Eτ(η′′_{k,j})) defined

with their help. First we claim that the sums T_{k} converge in distribution to an infinitely divisible distribution function whose characteristic function has a logarithm given by formula (1.3). Indeed, T_{k} = T′_{k} + T′′_{k}, the random variables T′_{k} and T′′_{k} are independent, and the random variables T′_{k} converge in distribution to the Gaussian distribution with expectation zero and variance M_{0}({0}) by Proposition 1. On the other hand, the random variables T′′_{k} converge in distribution to an infinitely divisible distribution determined by the canonical measure M′′_{0} because of Proposition 2, formula (1.9) and the convergence of the canonical measure M̄′′_{k} to the canonical measure M′′_{0}. (The measure M′′_{0} is the restriction of the measure M_{0} to the set R^{1} \ {0}.) The convergence of M̄′′_{k} to M′′ − 0 follows from the form of the measures M̄′′_{k}, the convergence of the measures M′′_{k} to

M′′ − 0 by Lemma 1 and the convergence of the measures ∑_{j=1}^{n_{k}} \frac{p̄_{k,j}−p_{k,j}}{p_{k,j}} G_{k,j}( du) to the

measure identically zero on the real line. The last convergence holds, since \frac{p̄_{k,j}−p_{k,j}}{p_{k,j}} ≥ 0,

and lim_{k→∞} sup_{1≤j≤n_{k}} \frac{p̄_{k,j}−p_{k,j}}{p_{k,j}} → 0. Indeed, by the identity 1 − e^{−p̄_{k,j}} = p_{k,j} defining the

quantity p̄_{k.j} we have p̄_{k,j} = − log(1 − p_{k,j}). Hence we have p_{k,j} ≤ p̄_{k,j} ≤ p_{k,j} + p_{k,j}^{2} for large indices k. (Here we exploit that for a large index k all numbers p_{k,j} are very small. Hence the relations formulated for the quantities p_{k,j} and p̄_{k,j} really hold.

Finally Proposition 1 and Theorem A enable us to prove Theorem 1, i.e. the state-

ment that the random sums S_{k} = ∑_{j=1}^{n_{k}} ξ_{k,j}, or what is equivalent to it, the random sums

S̃_{k} = ∑_{j=1}^{n_{k}} ξ̃_{k,j} = ∑_{j=1}^{n_{k}} (ξ̃_{k,j} − Eτ(ξ̃_{k,j})) of the random variables ξ̃_{k,j} defined in the cou-

pling construction described after Lemma 1 satisfy the statement of Theorem 1. Indeed, formula (1.8) implies that S̃_{k} − T_{k} ⇒ 0 where ⇒ denotes stochastic convergence. Hence the random variables S_{k} or S̃_{k} converge to the same distribution function as the random variables T_{k}, and this implies Theorem 1.

Let us make some comments about the proof of Theorem 1 explained in this paper. This approach also may explain that although Theorem 1 supplies many cases when the normalized sums of independent random variables have a limit distribution, the central limit theorem, i.e. the case when the limit is Gaussian deserves its name, the limit theorems with a Gaussian limit really play a central role in the theory of limit theorems for sums of independent random variables. Let us observe that in the coupling construction applied in this proof we approximated each term which contributes to the non-Gaussian part of the limit individually by a random variable with an infinitely divisible distribution. Then we showed the sum of the errors caused by these approximations is negligible. This fact can be interpreted in such a way that limit theorems with a non-Gaussian limit must have a very special form, in a certain sense the distributions of the terms in the sum must resemble to the limit distribution. The picture in the case of the central limit theorem is quite different. In this case, — and this is one of the most remarkable facts in probability theory — the distribution of the individual terms in the sum may be quite general, the distribution of the sum “forgets” the distribution of the individual terms. In such a case a term by term approximation of the summands independently of each other, — and this was done in the coupling construction for the non-Gaussian part — would cause a non-negligible error.

The method of proof given here and in Part II was quite different. Here we applied the so-called coupling method and proved the result by means of a probabilistic argument. In the proof of Part II the characteristic function technique, a useful method of analysis was applied. Nevertheless, it may be useful to understand that these two approaches are not so far from each other as it may seem at first sight. The coupling argument is also present in a hidden way also in the proof by means of characteristic functions.

Indeed, let us consider a triangular array ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k} which satisfies the uniform smallness condition. Let φ_{k,j}(t) denote the characteristic function

of the random variable ξ_{k,j}. Put S_{k} = ∑_{j=1}^{n_{k}} ξ_{k,j}. If the sums S_{k} have a limit distribution

then the relation

lim_{k→∞} ∏_{j=1}^{n_{k}} φ_{k,j}(t) = ψ(t)

holds with an appropriate characteristic function ψ(t). We have seen the (non-trivial) fact that in the last formula logarithm can be taken, i.e. this relation is equivalent to the formula

lim_{k→∞} ∑_{j=1}^{n_{k}} log φ_{k,j}(t) = log ψ(t)

Another important step of the proof was to show that since the random variables ξ_{k,j} are relatively small the replacement of the term log φ_{k,j}(t) by φ_{k,j}(t) − 1 is allowed, i.e. we can write

lim_{k→∞} ∑_{j=1}^{n_{k}}(φ_{k,j}(t) − 1) = log ψ(t).

Let us also observe that, as we have seen in Part I, the function φ_{k,j}(t) − 1 is the logarithm of the characteristic function of the infinitely divisible random variable which is determined by the Poisson process whose counting measure is the distribution function F_{k,j} of the random variable ξ_{k,j}. In such a way the replacement of the function log φ_{k,j}(t) by φ_{k,j}(t) − 1 corresponds to the coupling construction made in this part of the work.

Finally, we remark that the above coupling method enables us to approximate

not only the sums S_{k} = ∑_{j=1}^{n_{k}} ξ_{k,j} by means of random variables with infinitely divisible

distribution, but under some natural conditions the partial sums S_{k,l} = ∑_{j=1}^{l} ξ_{k,j}, 1 ≤

k ≤ n_{k}, can be approximated simultaneously by means of sums of independent random variables with indivisible distribution. This argument leads to the investigation of the so-called functional limit theorems, a result which deserves a more detailed discussion.

A.) Functional limit theorems for general triangular arrays.

An interesting and important result of probability theory, called the functional central limit theorem or invariance principle in the literature, states that if a triangular array satisfies the central limit theorem, then the broken line processes made from the partial sums of these random variables in a natural way converge weakly to a Wiener process. More explicitly, let ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, be a triangular array

such that Eξ_{k,j} = 0, k = 1, 2, …, 1 ≤ j ≤ n_{k}, lim_{k→∞} ∑_{j=1}^{n_{k}} Eξ_{j,k}^{2} = 1, and the Lindeberg

condition lim_{k→∞} ∑_{j=1}^{n_{k}} Eξ_{j,k}^{2}I(|ξ_{j.k}| > ε) = 0 holds for all ε > 0. Define the partial sums

S_{k,l} = ∑_{j=1}^{l} ξ_{k,j}, k = 1, 2, …, 1 ≤ l ≤ n_{k}, the numbers u_{k,0} = 0 and u_{k,l} = \frac{1}{D_{k}} ∑_{j=1}^{l} Eξ_{k,j}^{2},

1 ≤ l ≤ n_{k}, in the interval [0, 1] where D_{k} = ∑_{j=1}^{n_{k}} Eξ_{k,j}^{2}. Then we can define with

the help of these quantities the random broken lines S_{k}(t), 0 ≤ t ≤ 1, in such a way that S_{k}(0) = 0, S_{k}(u_{k,l}) = S_{k,l}, 1 ≤ l ≤ n_{k}, and the functions S_{k}(t) are linear in all intervals [u_{k,l−1}, u_{k,l}], 1 ≤ l ≤ n_{k}. The functional central limit theorem states that the distributions of the stochastic processes S_{k}(t), 0 ≤ t ≤ 1, considered as C([0, 1]) valued random variables, converge weakly to the distribution of a Wiener process. Let us emphasize that this result states in particular that the necessary and sufficient condition of the central limit theorem implies at the same time a stronger result.

The question arises whether the general limit theorems considered in this work have a similar functional limit version. There is a positive answer to this question. The limit theorem formulated in Theorem 1 holds if and only if an appropriately defined sequence of canonical measures on the real line converges to a canonical measure M_{0}. We shall show that we can introduce and define an appropriate sequence of canonical measures on the strip R^{1} × [0, 1] which are closely related to the canonical measures considered in Theorem 1, and their convergence implies a functional limit theorem for the appropriately defined random broken lines made from the sums of partial sums of the random variables in the triangular array. We shall formulate such a result in Theorem 2. But to do this first we have to introduce some definitions and notations.

To formulate Theorem 2 first we introduce the appropriate function space where we shall work. This space is called the D([0, 1]) space in the literature.

We say that a function x(t), 0 ≤ t ≤ 1, is a cadlag function (continue à droite, limite à gauche) if the function x(t) is continuous from the right, and it also has a left-hand side limit in all points. The space D([0, 1]) consists of the cadlag functions in the interval [0, 1], and an appropriate distance is introduced in it. A possible definition of this metric is the distance d(·, ·) defined in the following way: Let x, y ∈ D([0, 1]) be two cadlag functions and ε > 0 a real number. The relation d(x, y) ≤ ε holds if there exists a strictly monotone function λ(t) which is a homeomorphism of the interval [0, 1] into itself, sup_{0≤t≤1} |λ(t) − t| ≤ ε, and sup_{0≤t≤1} |y(t) − x(λ(t))| ≤ ε.

The space D([0, 1]) is a separable metric space with the above distance, but it is not a complete metric space. The property that two cadlag functions x(·) and y(·) are close to each other with respect to the metric d(·, ·) means that although these two functions may be far from each other with respect to the supremum norm, but they can put close to each other with respect to the this norm if the argument of one of these functions is slightly perturbed in an appropriate way. We introduced the space D([0, 1]) because we need this notion in the formulation of Theorem 2. Here we only formulate the results about this space but omit the proofs. All of them can be found in P. Billingsley’s book “Convergence of probability measures”. We had to introduce this notion, because the possible limit processes in Theorem 2, — the Poisson process is a typical example, — do not have continuous trajectories, hence we have to work in a different function space.

Let us remark that the above metric is not the only possible good metric which can be introduced in the space D([0, 1]). For instance the following metric d_{0}(·, ·) in the space D([0, 1]) is often applied in the literature. Let x(·), y(·) be two cadlag functions. We say that d_{0}(x, y) ≤ ε if there exists such a homeomorphism λ(·): [0, 1] → [0, 1] of the interval [0, 1] into itself for which λ(0) = 0, sup_{t≠s} log |\frac{λ(t)−λ(s)}{t−s} | ≤ ε, and |x(t)−y(λ(t))| ≤ ε for all

numbers t ∈ [0, 1]. It can be proved that the metrics d(·, ·) and d_{0}(·, ·) define the same topology on the space D([0, 1]). This means that sequences of probability measures on the space D([0, 1]) simultaneously converge or do not converge with respect to these metrics. Hence they are equivalent for our purposes. The essential difference between these two metrics is that the space D([0, 1]) is a separable complete metric space with respect to the metric d_{0}(·, ·), but this relation does not hold for the metric d_{0}(·, ·). Some proofs are simpler in complete metric space, and this is the main reason why the metric d_{0}(·, ·) is applied in several cases.

Let us remark that in the formulation of Theorem A we only assumed that the metric space (X, ρ) we have considered is separable, but did not demand that it has to be complete. This makes possible to apply the metric d(·, ·) in subsequent proofs, and this simplifies certain arguments.

Now we define the stochastic processes and canonical measures which will appear in Theorem 2. Let a triangular array ξ_{k,j}, 1 ≤ j ≤ n_{k}, be given, and let F_{k,j} denote the distribution function of the random variable ξ_{k,j}. Let us define the partial sums

S_{k,0} = 0, S_{k,l} = ∑_{j=1}^{l} ξ_{k,j}, 1 ≤ l ≤ n_{k}  (1.11)

Then let us fix for all numbers k = 1, 2, … an appropriate sequence of numbers 0 = u_{k,0} ≤ u_{k,1} ≤ u_{k,2} ≤ ⋯ ≤ u_{k,n_{k}} = 1 and define the random cadlag functions in the interval [0, 1] as

\begin{aligned}S_{k}(t) = S_{k}(t, u_{k,0}, …, u_{k,n_{k}}) & = S_{k,l−1}, if u_{k,l−1} ≤ t < u_{k,l}, 1 ≤ l ≤ n_{k}, \\ S_{k}(1) & = S_{k,n_{k}}, \end{aligned} (1.12)

k = 1, 2, …, with their help. (These numbers 0 = u_{k,0} ≤ u_{k,1} ≤ u_{k,2} ≤ ⋯ ≤ u_{k,n_{k}} = 1 are needed to define the appropriate scaling in the definition of the random cadlag functions. We cannot give them in such an explicit way as in the functional central limit theorem.) Furthermore, let us define certain σ-finite measures N_{k} on the direct product of the real line and the interval [0, 1], on the set R^{1} × [0, 1], in the following way: Let 0 = u_{k,0} ≤ u_{k,1} ≤ u_{k,2} ≤ ⋯ ≤ u_{k,n_{k}} = 1 be the same sequence of numbers which appeared in formula (1.12).

The measure N_{k}(·) is concentrated on the set R^{1} × ⋃_{l=1}^{n_{k}}{u_{k,l}},  (1.13)

and the restriction of the measures N_{k}(·) to the lines {(t, u): t ∈ R^{1}, u = u_{k,l}}, 1 ≤ l ≤ n_{k}, equals the measure x^{2}F_{k,l}(dx), i.e.

N_{k}(B × {u_{k,l}}) = ∫_{B} x^{2}F_{k,l}( dx), if 1 ≤ l ≤ n_{k}, and B ⊂ R^{1} is a measurable set.

(1.14)

Let us define the notion of canonical measures on the strip R^{1} × [0, 1] and their convergence. The measures N_{k} defined in formulas (1.13) and (1.14) also satisfy the properties of canonical measures.

The definition of canonical measures and their convergence. We call a σ-finite measure N(·) on the strip R^{1} × [0, 1] canonical if for all numbers s > 0

N([−s, s] × [0, 1]) < ∞, and ∫_{\{(u,v): |u|>s, 0≤v≤1\}} \frac{N( du, dv)}{u^{2}} < ∞.

Let a sequence N_{k}, k = 0, 1, 2, …, of canonical measures be given on the strip R^{1}×[0, 1]. We say that this sequence of canonical measures N_{k} converges (weakly) to a canonical measure N_{0} on the strip R^{1} × [0, 1] as k → ∞ if for all such numbers 0 ≤ a ≤ b ≤ 1 which are points of continuity of the limit measure N_{0}, i.e. for which lim_{ε→0} N_{0}([−R, R] ×

[a−ε, a+ε]) = 0, lim_{ε→0} N_{0}([−R, R]×[b−ε, b+ε]) = 0 and lim_{ε→0} ∫_{|u|>R, |v−a|<ε} \frac{N_{0}(du,dv)}{u^{2}} = 0,

lim_{ε→0} ∫_{|u|>R, |v−b|<ε} \frac{N_{0}(du,dv)}{u^{2}} = 0 for all numbers R > 0, the canonical measures M_{k,a,b},

k = 1, 2, …, on the real line defined by the formula M_{k,a,b}(B) = N_{k}(B × [a, b]) for all measurable sets B ∈ R^{1} converge to the canonical measure M_{0,a,b}(B) on the real line, where M_{0,a,b}(B) = N_{0}(B × [a, b]), for all measurable sets B ∈ R^{1}.

Now we formulate Theorem 2. It says that if the canonical measures N_{k} on the strip R^{1} × [0, 1] defined in formulas (1.13) and (1.14) converge to a canonical measure N_{0} on the strip R^{1} × [0, 1] then the stochastic processes S_{k}(·) defined in formula (1.12), considered as D([0, 1]) space valued random variables, converge weakly to a random process with cadlag trajectories which is determined by the limit canonical measure N_{0} in a natural way.

Theorem 2. Let ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ n_{k}, be a triangular array satisfying the uniform smallness condition and such that Eτ(ξ_{k,j}) = 0 far all k = 1, 2, …, 1 ≤ j ≤ n_{k} with the function τ(x) = τ_{a}(x) defined in formula (1.2) where a > 0 is some fixed number. Let us fix for all numbers k = 1, 2, … a sequence of numbers 0 = u_{k,0} ≤

u_{k,1} ≤ u_{k,2} ≤ ⋯ ≤ u_{k,n_{k}} = 1 satisfying the relation lim_{k→∞} sup_{1≤l≤n_{k}} |u_{k,l} − u_{k,l−1}| = 0,

and let us consider the canonical measures M_{k} defined on the strip R^{1} × [0, 1] defined by formulas (1.13) and (1.14) with the above sequences of numbers 0 = u_{k,0} ≤ u_{k,1} ≤ u_{k,2} ≤ ⋯ ≤ u_{k,n_{k}} = 1 and the distribution functions F_{k,l} of the random variables ξ_{k,l}, k = 1, 2, …, 1 ≤ l ≤ n_{k}. Let us assume that these canonical measures N_{k} on the strip R^{1} × [0, 1] converge weakly to a canonical measure N_{0} which satisfies the relation N_{0}(R^{1} × {0}) = 0. Furthermore, let us also assume that

a.) The function λ(t) = N_{0}({0} × [0, t]) defined with the help of the canonical measure

N_{0} is continuous in the interval [0, 1]. b.) For all numbers b > 0 the function

ν_{b}(t) = ∫_{\{(x,y): |x|>b, 0≤y≤t\}} \frac{N_{0}(dx, dy)}{x^{2}}, 0 ≤ t ≤ 1,

defined with the help of the limit canonical measure N_{0} is continuous in the interval [0, 1].

Then the stochastic processes S_{k}(t), 0 ≤ t ≤ 1, defined in formula (1.12) considered as D([0, 1]) valued random variables converge weakly to a stochastic process S(t), 0 ≤ t ≤ 1, which can be considered as a D([0, 1]) valued random variable. This process S(t) is a stochastic process with cadlag trajectories, it has independent increments, and it is determined by a Poisson field on R^{1} ×[0, 1] with counting measure \frac{N_{0}(du,dv)}{u^{2}} in a natural way. The distribution of this stochastic process can be described in the following way. We have S_{0} ≡ 0, and since S(t) is a process with independent increments, it is enough to give the distribution of the increments S(v) − S(u), 0 ≤ u ≤ v ≤ 1. The distribution function of such a difference is an infinitely divisible distribution determined by the canonical measure \frac{M_{0,u,v}(dx)}{x^{2}} with M_{0,u,v}(dx) = N_{0}(dx × (u, v]). The characteristic function φ_{u,v}(·) of the random variable S(v) − S(u) has a logarithm which is given by the following modified version of formula (1.3):

log φ_{u,v}(t) = ∫_{−∞}^{∞} \frac{e^{itx} − 1 − itτ(x)}{x^{2}}M_{0,u,v}( dx).  (1.15)

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

Remark 2. The conditions a.) and b.) imposed for the limit canonical measure N_{0} can be formulated in a unified way as N_{0}(R^{1} ×{t}) = 0 for all 0 ≤ t ≤ 1. This condition cannot be dropped. We show this with the help of an example explained in a rather sketchy way. Let us consider a triangular array ξ_{k,j}, k = 1, 2, …, 1 ≤ j ≤ k, (i.e. n_{k} = k) which is defined in the following way. Consider a sequence of numbers ε_{k}, k = 1, 2, …, such

that lim_{k→∞} ε_{k} = 0 and lim_{k→∞} kε_{k} = ∞. Let ξ_{k,j} = ζ_{k,j} + η_{k,j} if (\frac{1}{2} − 2ε_{k})k < j ≤ (\frac{1}{2} − ε_{k})k

or (\frac{1}{2} − 4ε_{k})k < j ≤ (\frac{1}{2} − 3ε_{k})k and ξ_{k,j} = ζ_{k,j} if j ∈ [1, k] \ (((\frac{1}{2} − 2ε_{k})k, \frac{1}{2} − ε_{k})k] ∪ ((\frac{1}{2} − 2ε_{k})k, \frac{1}{2} − ε_{k})k]) where the ζ_{k,j} are independent Gaussian random variables with expectation zero and variance \frac{1}{k}, and η_{k,j} are independent random variables, P (η_{k,j} = 1) = 1 − P (η_{k,j} = 0) = \frac{ε_{k}}{2k} which are also independent of the random variables ζ_{k,j}. Define the numbers u_{k,l} appearing in the formulation of Theorem 2 by the formula u_{k,l} = \frac{l}{k}, 0 ≤ l ≤ k. (Actually, it was not necessary to take a Gaussian part ζ_{k,j} in this example, we only introduced it to make the choice of the numbers u_{k,l} more natural.) It is not difficult to see that in this case the canonical measures N_{k} introduced before the formulation of Theorem 2 converge to the canonical measure N_{0} = N′_{0} + N′′_{0}, where N′_{0} is the Lebesgue measure on the set {0} × [0, 1], and the measure N′′_{0} is concentrated in the point (1, \frac{1}{2}), N′′_{0}(\{(1, \frac{1}{2})\}) = 1. We show that, as the limit measure N_{0} does not satisfies condition b.), the processes S_{k}(t) do not converge to the processes determined by the measure N_{0}. To see this observe that if we disregard the Gaussian part of this

example, i.e. we consider the partial sums S′′_{k,l} = ∑_{j=1}^{l} η_{k,j}, 1 ≤ l ≤ k, the stochastic

processes S′′_{k}(t) = S_{k,l} if u_{k,l} ≤ t < t_{k}, and the candidate for the limit process is the process determined by the measure N′′_{0}. This limit process S_{0}(t), 0 ≤ t ≤ 1, is defined as S_{0}(t) = 0 if 0 ≤ t < \frac{1}{2} and S_{0}(t) = η if \frac{1}{2} ≤ t ≤ 1 where η is a Poissonian random variable with parameter 1. The distributions of the processes S_{k}(·) do not converge to the distribution of the process S_{0}(·). Indeed, the random variables S_{k}(\frac{1}{2} − 4ε_{k}), S_{k}(\frac{1}{2} − 3ε_{k}), S_{k}(\frac{1}{2} − ε_{k}) take three different integer values with a positive probability, while the process S′′_{0}(·) can take at most two values with probability one. This excludes the convergence of the distributions of the processes S_{k}(·) to those of the process S_{0}(·) in the D([0, 1]) space. Since the Gaussian part of the processes S_{k}(·) have very small fluctuation in the interval [\frac{1}{2} −4ε_{k}, \frac{1}{2}] the original example also yields a counter-example.

The proof of Theorem 2 will be similar to that of Theorem 1. We can split the stochastic processes S_{k}(·) with the help of Proposition 1 and Lemma 1 to two parts, one of them responsible for the Gaussian the other one for the Poissonian part of the limit. The convergence to the Gaussian part of the limit follows from the functional central limit theorem. Then the converge to the Poissonian part of the limit process can be investigated by a refined version of the coupling argument applied in the proof of Theorem 1. Let us remark that the expression sup_{1≤j≤n_{k}} appeared in formula (1.8)

because it was appropriate in this form in the proof of Theorem 2.

2. The proof of Theorem 1.

In this section we prove the results applied in the proof of Theorem 1, Lemma 1 and Propositions 1 and 2.

The proof of Lemma 1. Let us choose a monotone decreasing sequence of positive numbers η_{p}, p = 1, 2, …, such that lim_{p→∞} η_{p} = 0 and the numbers ±η_{p} are points

of continuity of the measure M_{0}. Then lim_{p→∞} M_{0}((−η_{p}, η_{p})) = M_{0}({0}). Beside this,

there exists a threshold index k_{0}(p) for all numbers p = 1, 2, … such that for all k ≥ k_{0}(p) the inequality |M_{k}((−η_{p}, η_{p})) − M_{0}(−η_{p}, η_{p}))| ≤ \frac{1}{p} holds, and if the origin is a point of continuity of the measure M_{0}, i.e. M_{0}({0}) = 0, then also the inequality |M_{k}((0, η_{p}))−M_{0}(0, η_{p}))| ≤ \frac{1}{p} holds. We may also assume that |M_{0}^{±}(η_{p})−M_{k}^{±}(η_{p})| ≤ \frac{1}{p} if k ≥ k_{0}(p) where M_{0}^{±}(·) and the functions M_{k}^{±}(·) are the functions defined in formula (1.4). Because of the uniform smallness condition we can guarantee that

sup_{1≤j≤n_{k}}(1 − F_{k,j}(η_{p})) + sup_{1≤j≤n_{k}} F_{k,j}(−η_{p}) ≤ \frac{1}{p(M_{0}^{+}(η_{p}) + M_{0}^{−}(η_{p})) + 2}

if k ≥ k_{0}(p), and the threshold index k_{0}(p) is chosen sufficiently large. We may also assume that the sequence of threshold indices k_{0}(p), p = 1, 2, …, is monotone increasing. Put ε_{k} = η_{p} if k_{0}(p) ≤ k < k_{0}(p + 1). With such a choice of the numbers ε_{k} the statements of Lemma 1 hold. Indeed,

lim_{k→∞} M′_{k}([a, b]) = lim_{k→∞} M′_{k}([−η_{k_{0}(p)}, η_{k_{0}(p)}]) = M_{0}({0}) = M′_{0}([a, b])

if the interval [a, b] contains the origin in its interior, and lim_{k→∞} M′_{k}([a, b]) = 0 = M′_{0}([a, b])

if the interval [a, b] does not contain the origin in its interior and the points a and b are points of continuity of the measure M_{0} (in particular also in the case if 0 is a point of continuity of the measure M_{0}, and a = 0 or b = 0). These relations together with the fact that the measures M′_{k}, k = 1, 2, …, and M′_{0} are concentrated in a finite interval [−A, A] imply that the measures M′_{k} weakly converge to the measure M′_{0}. As the sequence of measures M_{k} converges (weakly) to the measure M_{0} and the sequence of measures M′_{k} converges (weakly) to the measure M′_{0} also the measures M′′_{k} = M_{k} − M′_{k} converge (weakly) to the measure M′′_{0} = M_{0} − M′_{0}. Furthermore,

∑_{j=1}^{n_{k}}[(1 − F_{k,j}(ε_{k})) + F_{k,j}(−ε_{k})]^{2}

≤ sup_{1≤j≤n_{k}}[(1 − F_{k,j}(ε_{k})) + F_{k,j}(−ε_{k})] ∑_{j=1}^{n_{k}}[(1 − F_{k,j}(ε_{k})) + F_{k,j}(−ε_{k})]

= sup_{1≤j≤n_{k}}[(1 − F_{k,j}(ε_{k})) + F_{k,j}(−ε_{k})](M_{k}^{+}(ε_{k}) + M_{k}^{−}(ε_{k}))

≤ \frac{1}{p(M_{0}^{+}(η_{p}) + M_{0}^{−}(η_{p})) + 2} (M_{0}^{+}(ε_{k}) + M_{0}^{−}(ε_{k}) + \frac{2}{p}) ≤ \frac{1}{p}

if k ≥ k_{0}(p). From here we get formula (1.5) by taking the limit procedure k → ∞.

Proof of Proposition 1. As ξ̃_{k,j} = η′_{k,j}I(ζ_{k,j} = 0) + γ_{k,j,1}I(ζ_{k,j} ≥ 1), and the random variables ζ_{k,j} are independent of the other random variables, hence

P (ξ̃_{k,j} ∈ A) = P (ξ̃_{k,j} ∈ A|ζ_{k,j} = 0)P (ζ_{k,j} = 0) + P (ξ̃_{k,j} ∈ A|ζ_{k,j} ≥ 1)P (ζ_{k,j} ≥ 1)
= P (η′_{k,j} ∈ A)P (ζ_{k,j} = 0) + P (γ_{k,j,1} ∈ A)P (ζ_{k,j} ≥ 1)
= ν̄(A)(1 − p_{k,j}) + ¯ ν̄(A)p_{k,j}
= P (ξ_{k,j} ∈ A ∩ {x: |x| < ε_{k}}) + P (ξ_{k,j} ∈ A ∩ {x: |x| ≥ ε_{k}})
= P (ξ_{k,j} ∈ A),

for all measurable sets A ⊂ R^{1}, that is the random variables ξ̃_{k,j} and ξ_{k,j} have the same distribution. The coupling construction also implies that for a fixed index k the random variables ξ̃_{k,j}, 1 ≤ j ≤ n_{k}, are independent. This construction also implies that for a fixed k the random variables η′_{k,j}, η′′_{k,j}, 1 ≤ j ≤ n_{k}, are independent, and

P (|ξ′_{k,j}| ≤ ε_{k}) = P (|η′_{k,j}| ≤ ε_{k}) = 1. To prove the relation lim_{k→∞} ∑_{j=1}^{n_{k}} |Eξ′_{k,j} − Eη′_{k,j}| = 0

let us observe that

|Eξ′_{k,j} − Eη′_{k,j}| = (1 − P (ζ_{k,j} = 0))|Eη′_{k,j}| = \frac{1 − P (ζ_{k,j} = 0)}{P (ζ_{k,j} = 0)}|Eξ′_{k,j}|

= \frac{P (|ξ_{k,j}| ≥ ε_{k})}{P (|ξ_{k,j}| < ε_{k})}|Eτ(ξ′′_{k,j})| ≤ 2P (|ξ_{k,j}| ≥ ε_{k})|Eτ(ξ′′_{k,j})|

if k ≥ k_{0} with an appropriate constant k_{0}, since 0 = Eτ(ξ̃_{k,j}) = Eξ′_{k,j} + Eτ(ξ′′_{k,j}), and P (ζ_{k,j} ≥ 1) = 1 − e^{−p̄_{k,j}} = p_{k,j} = P (|ξ_{k,j}| ≥ ε_{k}), where p̄_{k,j} is the solution of the equation p_{k,j} = 1 − e^{−p̄_{k,j}}, p_{k,j} = P (|ξ_{k,j}| ≥ ε_{k}), and p_{k,j} ≤ \frac{1}{2} if k ≥ k_{0}. As |Eτ(ξ′′_{k,j})| ≤ aP (|ξ_{k,j}| ≥ ε_{k}) this implies that |Eξ′_{k,j} − Eη′_{k,j}| ≤ 2aP (|ξ_{k,j}| ≤ ε_{k})^{2} = 2a [(1 − F_{k,j}(ε_{k})) + F_{k,j}(−ε_{k})]^{2}. We get relation (1.7) by summing up these inequalities and applying formula (1.5).

We can prove the first relation of formula (1.8) with the help of the Kolmogorov inequality. Indeed, for all numbers ε > 0

P ( sup_{1≤p≤n_{k}} |∑_{j=1}^{p}(ξ′_{k,j} − Eξ′_{k,j}) − (η′_{k,j} − Eη′_{k,j})| > ε) ≤ \frac{∑_{j=1}^{n_{k}} Var (ξ′_{k,j} − η′_{k,j})}{ε^{2}}

= \frac{∑_{j=1}^{n_{k}}(1 − P (ζ_{k,j} = 0))^{2}Var η′_{k,j}}{ε^{2}} ≤ \frac{1}{ε^{2}} sup_{1≤j≤n_{k}}(1 − P (ζ_{k,j} = 0))^{2} ∑_{j=1}^{n_{k}} Eη′_{k,j}^{2}

= \frac{1}{ε^{2}} M_{k}([−ε_{k}, ε_{k}]) sup_{1≤j≤n_{k}} P (|ξ_{k,j}| ≥ ε_{k})^{2} → 0 if k → ∞,

because lim sup_{k→∞} M_{k}([−ε_{k}, ε_{k}]) ≤ M_{0}({0}) < ∞ in the construction of Lemma 1, and

sup_{1≤j≤n_{k}}(P (|ξ_{k,j}| ≥ ε_{k})^{2} → 0 if k → ∞.

To prove the second statement of formula (1.8) first we show that

Eτ(η′′_{k,j}) = \frac{p̄_{k,j}}{p_{k,j}} ∫_{\{|u|≥ε_{k}\}} τ(u)F_{k,j}( du) = \frac{p̄_{k,j}}{p_{k,j}} Eτ(ξ′′_{k,j}),  (2.1)

where p_{k,j} = P (|ξ_{k,j}| ≥ ε_{k}) = [(1 − F_{k,j}(ε_{k}) + F_{k,j}(−ε_{k})], and p̄_{k,j} is the solution of the equation 1 − e^{−p̄_{k,j}} = p_{k,j}. Indeed, by exploiting that if η_{1}, η_{2}, …, are independent, identically distributed random variables, τ is a random variable taking non-negative integer values which is independent of the random variables η_{j}, then E(η_{1} + ⋯ + η_{τ}) = EτEη_{1}. Further, since Eζ_{k,j} = p̄_{k,j} = 1 − e^{−p_{k,j}} we get that

E (∑_{l=1}^{ζ_{k,j}} γ_{k,j,l}I(|γ_{k,j,l}| ≤ a)) = \frac{Eζ_{k,j}}{p_{k,j}} ∫_{\{ε_{k}≤|u|≤a\}} τ(u)F_{k,j}( du)

= \frac{p̄_{k,j}}{p_{k,j}} ∫_{\{ε_{k}≤|u|≤a\}} τ(u)F_{k,j}( du),

and similarly

E (∑_{l=1}^{ζ_{k,j}} I(|γ_{k,j,l}| ≥ a)) = ¯ ν̄_{k,j}((−∞, −a] ∪ [a, ∞))Eζ_{k,j}

= \frac{p̄_{k,j}}{p_{k,j}} [1 − F_{k,j}(a) + F (−a_{k,j})] .

Since

Eτ(η′′_{k,j}) = E (∑_{l=1}^{ζ_{k,j}} γ_{k,j,l}I(|γ_{k,j,l}| ≤ a)) + aE (∑_{l=1}^{ζ_{k,j}} I(|γ_{k,j,l}| ≥ a)) ,

the above two identities imply formula (2.1).

To prove the second relation of formula (1.8) let us also observe that

ξ′′_{k,j} − η′′_{k,j} = ξ′′_{k,j} − ∑_{l=1}^{ζ_{k,j}} γ_{k,j,l} = I(ζ_{k,j} ≥ 2) ∑_{l=2}^{ζ_{k,j}} γ_{k,j,l},

because by the coupling construction ξ′′_{k,j} = η′′_{k,j} = 0, on the set {ω: ζ_{k,j}(ω) = 0} and ξ′′_{k,j} = η′′_{k,j} = γ_{k,j,1} on the set {ω: ζ_{k,j}(ω) ≥ 1}. With the help of these relations we get that

P (∑_{j=1}^{p} ξ′′_{k,j} − ∑_{j=1}^{p} η′′_{k,j} ≠ 0 for some number 1 ≤ p ≤ n_{k}) ≤ ∑_{j=1}^{n_{k}} P (ζ_{k,j} ≥ 2)

≤ ∑_{j=1}^{n_{k}} [(1 − F_{k,j}(ε_{k}) + F_{k,j}(−ε_{k}))]^{2} → 0 if k → ∞

by the formula (1.5). Hence to prove the second relation of formula (1.8) it is enough to show that

lim_{k→∞} ∑_{j=1}^{n_{k}} |Eτ(ξ′′_{k,j}) − Eτ(η′′_{k,j})| = 0.

But by formula (2.1)

|Eτ(ξ′′_{k,j}) − Eτ(η′′_{k,j})| = |\frac{p̄_{k,j} − p_{k,j}}{p_{k,j}} Eτ(ξ′′_{k,j})| ≤ 2ap_{k,j}^{2} = 2aP^{2}(|ξ_{k,j}| ≥ ε_{k})

if k ≥ k_{0}. (Observe that |1 − e^{−p_{k,j}} − p_{k,j}| < p_{k,j}^{2} and |Eτ(ξ′′_{k,j}| ≤ 2ap_{k,j}.) This implies that

∑_{j=1}^{n_{k}} |Eτ(ξ′′_{k,j}) − Eτ(η′′_{k,j})| ≤ 2a ∑_{j=1}^{n_{k}} [(1 − F_{k,j}(ε_{k}) + F_{k,j}(−ε_{k})]^{2} → 0 if k → ∞

by formula (1.5). The above relations imply also the second part of formula (1.8).

Finally, we get by summing up the identities in formula (1.6) for the numbers 1 ≤ j ≤ n_{k} for a fixed integer k and by applying the definition given in Lemma 1 that

log E exp \{it (∑_{j=1}^{n_{k}}(η′′_{k,j})\} = ∫ \frac{e^{itu} − 1}{u^{2}} M̄′′_{k}( du),

and athe summation of the identity (2.1) in the variable j yields the identity

∑_{j=1}^{n_{k}} Eτ(η′′_{k,j})) = ∫ \frac{τ(u)}{u^{2}} M̄′′_{k}( du).

These relations imply formula (1.9). The random variables η′_{k,j} − Eη′_{k,j}, 1 ≤ j ≤ n_{k},

are independent, |η′_{k} − Eη′_{k}| ≤ ε_{k}, hence the sums T′_{k} = ∑_{j=1}^{n_{k}}(η′_{k,j} − Eη_{k,j}) satisfy the

central limit theorem. Beside this, ET′_{k} = 0, and we complete the proof of Lemma 1 if

we show that lim_{k→∞} ∑_{j=1}^{n_{k}} Var η′_{k,j} = M_{0}({0}). This follows from the identity

lim_{k→∞} ∑_{j=1}^{n_{k}} (Eη′_{k,j})^{2} = 0, lim_{k→∞} ∑_{j=1}^{n_{k}} |Eη′_{k,j}^{2} − Eξ′_{k,j}^{2}| = 0  (2.2)

to be proved below, since lim_{k→∞} ∑_{j=1}^{n_{k}} Eξ′_{k,j}^{2} = M_{0}({0}).

We get similarly to the proof of formula (1.7) that

|Eη′_{k,j}| = \frac{|Eξ′_{k,j}|}{P (ζ_{k,j} = 0)} = \frac{|Eτ(ξ′′_{k,j})|}{P (|ξ_{k,j}| < ε_{k})} ≤ 2|Eτ(ξ′′_{k,j})| ≤ 2aP (|ξ_{k,j}| > ε_{k})

if k ≥ k_{0} with an appropriate constant k_{0}. Then formula (1.5) implies the first statement of formula (2.2). On the other hand,

|Eξ′_{k,j}^{2} − Eη′_{k,j}^{2}| = (1 − P (ζ_{k,j} = 0))Eη′_{k,j}^{2} = \frac{1 − P (ζ_{k,j} = 0)}{P (ζ_{k,j} = 0)}Eξ′_{k,j}^{2}
≤ 2P (|ξ_{k,j}| ≥ ε_{k})Eξ′_{k,j}^{2}.

This inequality together with the Schwarz inequality imply that

∑_{j=1}^{n_{k}} |Eξ′_{k,j}^{2} − Eη′_{k,j}^{2}| ≤ (∑_{j=1}^{n_{k}} 4P^{2}(|ξ_{k,j}| ≥ ε_{k}) · ∑_{j=1}^{n_{k}} (Eξ′_{k,j}^{2})^{2})^{1/2} → 0, if k → ∞,

since lim_{k→∞} ∑_{j=1}^{n_{k}} P^{2}(|ξ_{k,j}| ≥ ε_{k}) = 0 by formula (1.5), and

∑_{j=1}^{n_{k}} (Eξ′_{k,j}^{2})^{2} ≤ const. ∑_{j=1}^{n_{k}} Eξ′_{k,j}^{2} ≤ const.

with an appropriate constant for all numbers k ≥ 1.

The proof of Proposition 2. As the canonical measures M_{k} converge to the canonical measure M_{0} and M_{0}({0}) = 0 for all numbers ε > 0 there exist such numbers δ = δ(ε) > 0, R = R(ε) and threshold index n̄ = n̄(ε) for which

M_{k}((−δ, δ)) < ε^{3}, and ∫_{\{u: |u|>R\}} \frac{1}{u^{2}} M_{k}( du) < ε if k ≥ n̄(ε).  (2.3)

We may also assume that the numbers ±δ = ±δ(ε) and ±R = ±R(ε) are points of continuity of the measure M_{0}.

Let us introduce the measures μ_{k}( dx) = \frac{M_{k}(dx)}{x^{2}}, k = 1, 2, … and μ_{0}( dx) = \frac{M_{0}(dx)}{x^{2}} on the real line. Let us choose such numbers δ = x_{1} < x_{2} < ⋯ < x_{s} = R with an appropriate index s for which ±x_{l} are points of continuity of the measure M_{0}, 1 ≤ l ≤ s,

and \frac{ε^{4}}{2L} < x_{l} − x_{l−1} < \frac{ε^{4}}{L}, 1 < l ≤ s with L = sup_{k≥0} μ_{k} ((−R, −δ) ∪ (δ, R)). Actually the

above defined sequence δ = x_{1} < x_{2} < ⋯ < x_{s} = R also depends on the number ε although we have not indicated this dependence.

Let us consider the sequence of numbers ε_{j} = 2^{−j}, j = 1, 2, …, We shall choose an appropriate sequence of numbers n_{j} = n_{j}(ε_{j}) ≥ n̄(ε_{j}), j = 1, 2, … and construct the random variables η_{k} and η̄_{k} with the help of the same sequence of numbers δ = x_{1} < x_{2} < ⋯ < x_{s} = R considered in the previous paragraph (which depends on ε = ε_{j}) for all indices n_{j} ≤ k ≤ n_{j+1}. We shall show that in the case of a good choice of the sequence n_{j} and a good construction of the random variables η_{k} and η̄_{k}, k = 1, 2, … we can satisfy the statement Proposition 2.

We shall construct the random variables η_{k} and η̄_{k} with infinitely divisible distributions by first constructing two Poisson processes ξ_{k,1}, ξ_{k,2}, … and ξ̄_{k,2}, ξ̄_{k,2}, … with counting measures μ_{k} and μ_{0} respectively. Then we define the random variables η_{k} and η̄_{k} as the regularized sums of these Poisson processes described in Part I. More explicitly, we put

\begin{aligned}η_{k}(ω) & = lim_{N→∞} (  ∑_{p: |ξ_{k,p}(ω)|≥2^{−N}} ξ_{k,p}(ω) − E (  ∑_{p: |ξ_{k,p}(ω)|≥2^{−N}} τ(ξ_{k,p}(ω)))) , \\ η̄_{k}(ω) & = lim_{N→∞} (  ∑_{p: |ξ̄_{k,p}(ω)|≥2^{−N}} ξ̄_{k,p}(ω) − E (  ∑_{p: |ξ̄_{k,p}(ω)|≥2^{−N}} τ(ξ̄_{k,p}(ω)))) , \end{aligned} (2.4)

where the function τ(x) = τ_{a}(x) was defined in formula (1.2). In Part I we have seen that the limits in formula (2,4) exist with probability 1, and the random variables η_{k} and η̄_{k} they define have the prescribed distributions.

To construct the Poisson processes ξ_{k,1}, ξ_{k,2}, … and ξ̄_{k,2}, ξ̄_{k,2}, … with counting measures μ_{k} and μ_{0} respectively first we construct some Poisson distributed random variables ζ_{k,l}^{±} and ζ̄_{k,l}^{±}, 1 ≤ l < s, from which the random variable ζ_{k,l}^{+} and ζ̄_{k,l}^{+}. l = 1, 2, …, tell us that the Poisson processes ξ_{k,n} and ξ̄_{k,n}, n = 1, 2, …, how many points have in the interval [x_{l}, x_{l+1}). Similarly, the random variables a ζ_{k,l}^{−} and ζ̄_{k,l}^{−} tell that these Poisson processes how many points have in the intervals [−x_{l+1}, −x_{l}).

To construct the above random variables and ζ_{k,l}^{±} and ζ̄_{k,l}^{±} let us first define independent Poisson random variables α_{k,l}^{±}, β_{k,l}^{±}, 1 ≤ l < s, with Poisson distribution such that the distribution of α_{k,l}^{+} has parameter min(μ_{k}((x_{l}, x_{l+1})), μ((x_{l}, x_{l+1}))) and the distribution of β_{k,l}^{+} has parameter

max(μ_{k}((x_{l}, x_{l+1})), μ_{0}((x_{l}, x_{l_{1}}))) − min(μ_{k}((x_{l}, x_{l+1}), μ_{0}(x_{l}, x_{l+1}))).

Similarly, let the distribution of α_{k,l}^{−} have parameter

min(μ_{k}((−x_{l+1}, −x_{l})), μ_{0}((−x_{l+1}, −x_{l}))),

and let the distribution of β_{k,l}^{−} have parameter

max(μ_{k}((−x_{l+1}, −x_{l})), μ_{0}((−x_{l+1}, −x_{l})))
− min(μ_{k}((−x_{l+1}, −x_{l})), μ_{0}((−x_{l+1}, −x_{l}))).

If μ_{k}((x_{l}, x_{l+1})) ≤ μ_{0}((x_{l}, x_{l+1})), then put ζ_{k,l}^{+} = α_{k,l}^{+}, ζ̄_{k,l}^{+} = α_{k,l}^{+} + β_{k,l}^{+}, and if μ_{k}((x_{l}, x_{l+1})) > μ_{0}((x_{l}, x_{l+1})), then put ζ̄_{k,l}^{+} = α_{k,l}^{+}, ζ_{k,l}^{+} = α_{k,l}^{+} + β_{k,l}^{+}, 1 ≤ l < s. Let us define similarly the random variables ζ_{k,l}^{−} and ζ̄_{k,l}^{−} only in this case we replace the interval (x_{l}, x_{l+1}) by the interval (−x_{l+1}, −x_{l}) and the random variables α_{k,l}^{+} and β_{k,l}^{+} by the random variables α_{k,l}^{−} and β_{k,l}^{−}.

The random variables ζ_{k,l}^{+} and ζ_{k,l}^{−} are Poisson distributed, and their parameters are μ_{k}((x_{l}, x_{l+1})) and μ_{k}((−x_{l+1}, −x_{l})). Similarly, the random variables ζ̄_{k,l}^{+} and ζ̄_{k,l}^{−} are Poisson distributed with parameters μ_{0}((x_{l}, x_{l+1})) and μ_{0}((−x_{l+1}, −x_{l})). This means that the distributions of the above constructed random variables agree with the distributions of the number of points in the appropriate intervals of the Poisson processes to be constructed. This property makes possible the application of these random variables in the construction. Let us also observe that these random variables also satisfy the identity

\begin{aligned}E |ζ_{k,l}^{+} − ζ̄_{k,l}^{+}| = |E (ζ_{k,l}^{+} − ζ̄_{k,l}^{+})| & = |μ_{k}((x_{l}, x_{l+1})) − μ_{k}((x_{l}, x_{l+1}))| , \\ E |ζ_{k,l}^{−} − ζ̄_{k,l}^{−}| = |E (ζ_{k,l}^{−} − ζ̄_{k,l}^{−})| & = |μ_{k}((−x_{l+1}, −x_{l})) − μ_{k}((−x_{l+1}, −x_{l}))| . \end{aligned} (2.5)

Now we turn to the construction of the Poisson processes ξ_{k,n} and ξ̄_{k,n}, n = 1, 2, …, with the help of the above constructed random variables ζ_{k,l}^{±} and ζ̄_{k,l}^{±}. Let us throw ζ_{k,l}^{+} number of points to the interval (x_{l}, x_{l+1}) and ζ_{k,l}^{−} number of points to the interval (−x_{l+1}, x_{l}) independently of each other so that these points fall into a set A ⊂ (x_{l}, x_{l+1}) or A ⊂ (x_{l}, x_{l+1}) with probability \frac{μ(A)}{μ((x_{l},x_{l+1}))} and \frac{μ(A)}{μ((x_{−l+1},−x_{l}))}, 1 ≤ l ≤ s, respectively. Similarly, let us consider the Poisson distributed random variable ζ_{k,0} with parameter μ_{k}(−x_{1}, x_{1}) and the Poisson distributed random variable ζ_{k,s}^{+} + ζ_{k,s}^{−} with parameter μ_{k}((−∞, −x_{s})∪(x_{s}, ∞)), and let us throw ζ_{k,0} number of points to the interval (−x_{1}, x_{1}) so that a point falls into a set A ⊂ (−x_{1}, x_{1}) with probability \frac{μ_{k}(A)}{μ_{k}((−x_{1},x_{1}))}, and let us throw ζ_{k,s}^{+} + ζ_{k,s}^{−} number of points to the set (−∞, −x_{s}) ∪ (x_{s}, ∞) so that a point falls into a set A ⊂ (−∞, −x_{s}) ∪ (x_{s}, ∞) with probability \frac{μ_{k}(A)}{μ_{k}((−∞,−x_{s})∪(x_{s},∞))}. Let the above considered random variables ζ_{k,l} together with all point throws made in the above construction be independent of each other. Then the union of the points thrown to different intervals is a Poisson process ξ_{k,1}, ξ_{k,2}, … with counting measure μ_{k}. We can construct similarly a Poisson process ξ̄_{k,1}, ξ̄_{k,2}, … with counting measure μ_{0} on the real line. Only in this case we replace the Poisson distributed random variables ζ_{k,l}^{±}, 0 ≤ l ≤ s, by the Poisson distributed random variables ζ̄_{k,l}^{±}, 0 ≤ l ≤ s, whose parameters can be given similarly, only the measure μ_{k} is replaced by the measure μ_{0}.

In such a way we have constructed the underlying Poisson processes and the random variables η_{k} and η̄_{k} determined by them. (Only the threshold index n_{j} = n_{j}(ε_{j}) is still not fixed.) We want to show that the above construction satisfies the stochastic convergence η_{k} − η̄_{k} ⇒ 0. The proof of this statement will be based on the observations that the underlying Poisson processes have almost the same number of points in the intervals (x_{l−1}, x_{l}), (the relation (2.5) expresses such a fact). Beside this, these intervals are very small, hence the precise position of the points falling to them has a very small influence on the value of the random variables η_{k} and η̄_{k}. To simplify further notations let us denote by B_{l} the interval (x_{l}, x_{l+1}) if 1 ≤ l < s and the interval (x_{l−1}, x_{l}) if −1 ≥ l > −s.

First we consider the contribution of those points of the Poisson processes to the random variables η_{k} and η̄_{k} which take a large value, more explicitly whose absolute values are larger than R with the number R introduced in formula (2.3). If k ≥ n_{j} ≥ n̄(ε_{j}), then the measure M_{k} satisfies relation (2.3) with ε_{j} = 2^{−j}. This relation also holds if the measures M_{k} are replaced by the limit measure M_{0}. By the second relation of formula (2.3) the parameters of the Poisson distributed random variables ζ_{k,s} = ζ_{k,s}^{+} + ζ_{k,s}^{−} and ζ̄_{k,s} = ζ̄_{k,s}^{+} + ζ̄_{k,s}^{−} are less than 2^{−j}. Hence the probability of the event that the corresponding Poisson processes ξ_{k,n} and ξ̄_{k,n}, n = 1, 2, …, contain no point such that |ξ_{k,n}| > R or |ξ̄_{k,n}| > R is greater than 1 − 2 · 2^{−j}. Furthermore, the expected number of the points of the Poisson processes with absolute value larger than R is less than 2 · 2^{−j}, and |τ(x)| ≤ a for all points x ∈ R^{1}. The above relations imply that

P (| ∑_{{n: |ξ_{k,n}|≥R}} ξ_{k,n} − E ( ∑_{{n: |ξ_{k,n}|≥R}} τ(ξ_{k,n}))| ≤ 2a · 2^{−j}) < 2 · 2^{−j} if k ≥ n_{j},

P (| ∑_{{n: |ξ̄_{k,n}|≥R}} ξ̄_{k,n} − E ( ∑_{{n: |ξ̄_{k,n}|≥R}} τ(ξ̄_{k,n}))| ≤ 2a · 2^{−j}) < 2 · 2^{−j} if k ≥ n_{j}. (2.6)

For a Poisson process ξ_{n}, n = 1, 2, …, in the interval [a, b] with a finite counting measure μ the variance of the random sum ∑_{n} ξ_{n} equals ∫_{a}^{b} u^{2}μ( du), (see e.g. Lemma 1

in Part I.) Hence the Chebishev inequality and the first part formula (2.3) imply that for all sufficiently large δ > 0 (observe that τ(x) = τ_{a}(x) = x if |x| ≤ δ.)

P (|  ∑_{{l: 2^{−N} ≤|ξ_{k,n}|≤δ}} ξ_{k,n} − E (  ∑_{{l: 2^{−N} ≤|ξ_{k,n}|≤δ}} τ(ξ_{k,n}))| ≥ 2^{−j})

≤ 2^{2j}Var ( ∑_{{l: 2^{−N} ≤|ξ_{k,p}|≤δ}} ξ_{k,n}) ≤ 2^{2j}ε_{j}^{3} = 2^{−j}

P (|  ∑_{{l: 2^{−N} ≤|ξ̄_{k,n}|≤δ}} ξ̄_{k,n} − E (  ∑_{{l: 2^{−N} ≤|ξ̄_{k,n}|≤δ}} τ(ξ̄_{k,n}))| ≥ 2^{−j})

≤ 2^{2j}Var ( ∑_{{n: 2^{−N} ≤|ξ̄_{k,n}|≤δ}} ξ̄_{k,n}) ≤ 2^{2j}ε_{j}^{3} = 2^{−j}

if k ≥ n_{j} and 2^{−N} < δ.  (2.7)

Further we claim that if the indices n_{j} are chosen sufficiently large, then

P ( ∑_{l: 1≤|l|<s} ( ∑_{n: ξ_{k,n}∈B_{l}} ξ_{k,n} − E ( ∑_{n: ξ_{k,p}∈B_{l}} τ(ξ_{k,n})))  (2.8)

− ( ∑_{n: ξ_{k,n}∈B_{l}} ξ̄_{k,n} − E ( ∑_{p: ξ_{k,n}∈B_{l}} τ(ξ̄_{k,n}))) > 2^{−j}) < 2^{−j} if k ≥ n_{j}

First we show with the help of formulas (2.4), (2.6), (2.7) and (2.8) that η_{k}−η̄_{k} ⇒ 0. Indeed, by summing up formulas (2.6), (2.7) and (2.8) we get that of all integers N such that 2^{−N} < δ

P ( ( ∑_{n: |ξ_{k,n}|≥2^{−N}} ξ_{k,n} − E ( ∑_{n: |ξ_{k,n}|≥2^{−N}} τ(ξ_{k,n})))  (2.9)

− ( ∑_{n: |ξ̄_{k,n}|≥2^{−N}} ξ̄_{k,n} − E ( ∑_{n: |ξ_{k,n}|≥2^{−N}} τ(ξ̄_{k,n}))) > (4a + 3) · 2^{−j}) < 5 · 2^{−j}

if k ≥ n_{j}.

Let us consider the lim inf of the events whose probabilities were investigated in

formula (2.9) in the variable N. (We recall that lim inf_{N→∞} A_{N} = ⋃_{N=1}^{∞} ( ⋂_{L=N}^{∞} A_{L}).) By

applying formula (2.4) we get that P (|η_{k} − η̄_{k}| > (4a + 3) · 2^{−j}) < 5 · 2^{−j} if k ≥ n_{j}. Hence formula(2.8) implies the relation η_{k} − η̄_{k} ⇒ 0 as we claimed. To give the still missing proof of formula (2.8) first we show that

P ( ∑_{l: 1≤|l|<s}( ∑_{n: ξ_{k,n}∈B_{l}}ξ_{k,n} − E ( ∑_{n: ξ_{k,n}∈B_{l}}τ(ξ_{k,n})) − ζ_{k,l}x_{l} + τ(x_{l})μ_{k}(B_{l})) > 2^{−2j})
< 2 · 2^{−2j} and
P ( ∑_{l: 1≤|l|<s}( ∑_{n: ξ̄_{k,n}∈B_{l}}ξ̄_{k,n} − E ( ∑_{n: ξ_{k,n}∈B_{l}}τ(ξ̄_{k,n})) − ζ̄_{k,l}x_{l} + τ(x_{l})μ_{0}(B_{l})) > 2^{−2j})
< 2 · 2^{−2j}

(2.10)

if k ≥ n_{j}.

The first inequality of formula (2.10) bounds the error we commit if those points of the Poisson processes ξ_{k,n}, n = 1, 2, …, for which ξ_{k,n} ∈ B_{l} are replaced by the endpoint x_{l} of the interval B_{l}, and sum up these errors for all such points of the Poisson process for which δ ≤ |ξ_{k,l}| ≤ R. As

|ξ_{k,n} − x_{l}| ≤ sup_{1≤|l|≤s} |x_{l+1} − x_{l}| ≤ \frac{ε_{j}^{4}}{L} ,

|E ( ∑_{n: ξ_{k,n}∈B_{l}} τ(ξ_{k,n}^{(l)})) − τ(x_{l})μ_{k}(B_{l})| ≤ sup_{1≤|l|≤s} |x_{l+1} − x_{l}|μ_{k}(B_{l}) ≤ \frac{ε_{j}^{4}}{L} μ_{k}(B_{l})

if k ≥ n_{j}, the first inequality of relation (2.10) follows from the relations

∑_{l: 1≤|l|<s} |E ( ∑_{n: ξ_{k,n}∈B_{l}} τ(ξ_{k,n}^{(l)})) − τ(x_{l})μ_{k}(B_{l})|

≤ ∑_{l: 1≤|l|<s} \frac{ε_{j}^{4}}{L} μ_{k}(B_{l}) = ε_{j}^{4} \frac{μ_{k}((−R, R) \ (−δ, δ))}{L} ≤ ε_{j}^{4}

and

P ( ∑_{l: 1≤|l|<s} | ∑_{n: ξ_{k,n}∈B_{l}} ξ_{k,n} − ζ_{k,l}x_{l}| > ε_{j}^{2})

≤ P (#\{n: δ < |ξ_{k,n}| < R\} > \frac{L}{2ε_{j}^{2}} ) ≤ \frac{2ε_{j}^{2}E (#\{n: δ < |ξ_{k,n}| < R\})}{L}

= \frac{2ε_{j}^{2}μ_{k} ((−R, −δ) ∪ (δ, R))}{2L} ≤ 2ε_{j}^{2} = 2 · 2^{−2j} if k ≥ n_{j}.

The second inequality of formula (2.10) can be proved similarly, only in this case we have to consider the Poisson process ξ̄_{k,n}, n = 1, 2, …, instead of the Poisson process ξ_{k,n}, n = 1, 2, …, and the measure μ_{k} has to be replaced by the measure μ_{0}.

By formula (2.10) to complete the proof of formula (2.8) hence of Proposition 2 it is enough to show that if the threshold indices n_{j} ≥ n̄_{j} are chosen sufficiently large, then

P ( ∑_{l: 1≤|l|<s} |x_{l}(ζ_{k,l} − ζ̄_{k,l}) − τ(x_{l})(μ_{k}(B_{l}) − μ_{0}(B_{l}))| > 2^{−2j}) ≤ \frac{2^{−j}}{2} if k ≥ n_{j}

(2.11)

or to prove the following slightly stronger inequality:

∑_{l: 1≤|l|<s} (|x_{l}| E|ζ_{k,l} − ζ̄_{k,l}| + |τ(x_{l})| |μ_{k}(B_{l}) − μ_{0}(B_{l})|) < 2^{−4j} if k ≥ n_{j}. (2.12)

But formula (2.5) implies that E|ζ_{k,l} − ζ̄_{k,l}| = C_{j}|μ_{k}(B_{l}) − μ_{0}(B_{l})|. Beside this, the identities lim_{k→∞} μ_{k}(B_{l}) = μ_{0}(B_{l}) hold because of the (weak) convergence of the canonical

measures M_{k} and M_{0}. As the sum in formula (2.12) contains only finitely many terms (the number of terms depends only on the index j), and each term tends to zero as k → ∞ by the above observations the expression at the left-hand side of formula (2.12) can be made an arbitrary small positive number by choosing the threshold index n_{j} sufficiently large.

3. The functional limit theorem. The proof of Theorem 2.

First we shall show with the help of Lemma 1 and Proposition 1 that also the stochastic processes S_{k}(t), 0 ≤ t ≤ 1, appearing in Theorem 2 can be split to two parts, one of them responsible for the convergence of the Gaussian and one of them responsible for the convergence of the Poissonian part. The convergence of the Gaussian part can be deduced from the functional central limit theorem and the convergence of the Poissonian part can be reduced to a simpler statement about the convergence of infinitely divisible processes. This will be the content of Part A in Section 3. In Part B we prove the convergence of the Poissonian part with the help of two Propositions. The first of them, Proposition 3, enables us to discretize the time parameter of the stochastic processes we investigate. Proposition 4 gives a good coupling of infinitely divisible processes whose canonical measures (on the strip R^{1} ×[0, 1]) are close to each other. It can be considered as a generalization of Theorem 2 where the coupling of infinitely divisible processes is considered instead of the coupling of infinitely divisible random variables. Finally in Part C we prove Propositions 3 and 4, and this completes the proof of Theorem 2.

A.) Poisson approximation. Separation of the normal and Poisson part of the limit process.

We reduce the proof of Theorem 2 to that of a simpler statement with the help of Lemma 1, Proposition 1 and the coupling construction described after Lemma 1.

Let us consider the random variables ξ′_{k,j}, ξ′′_{k,j}, η′_{k,j}, η′′_{k,j}, ξ̃_{k,j} = ξ′_{k,j} + ξ′′_{k,j} and η_{k,j} = η′_{k,j} + η′′_{k,j}, 1 ≤ j ≤ n_{k}, introduced in the coupling construction described after

Lemma 1 together with the partial sums S′_{k,l} = ∑_{j=1}^{l} ξ′_{k,j}, S′′_{k,l} = ∑_{j=1}^{l} ξ′′_{k,j}, S̃_{k,l} = S′_{k,l}+S′′_{k,l},

T′_{k,l} = ∑_{j=1}^{l}(η′_{k,j} − Eη′_{k,j}), T′′_{k,l} = ∑_{j=1}^{l}(η′′_{k,j} − Eτ(η′′_{k,j})), T_{k,l} = T′_{k,l} + T′′_{k,l}, 1 ≤ l ≤ n_{k}. Let

us also introduce the stochastic processes S′_{k}(t), S′′_{k}(t), S̃_{k}(t), T′_{k}(t), T′′_{k}(t) and T_{k}(t), 0 ≤ t ≤ 1, with cadlag function trajectories which we define similarly to the stochastic process S(t), 0 ≤ t ≤ 1, introduced in formula (1.12) with the difference that we replace the random variables S_{k,l} in formula (1.12) by the random variables S′_{k,l}, S′′_{k,l}, S̃_{k,l} and T′_{k,l}, T′′_{k,l}, T_{k,l}. It follows from formula (1.8) and the identity Eξ′_{k,j} + Eτ(ξ′′_{k,j}) = Eτ(ξ̃_{k,j}) = 0 that

sup_{0≤t≤1} |T_{k}(t) − S̃_{k}(t)| ⇒ 0, if k → ∞,  (3.1)

where ⇒ denotes stochastic convergence. The distributions of the stochastic processes S_{k}(t) and S̃_{k}(t) agree. Furthermore, if x_{k}(·) and y_{k}(·) are such functions in the space D([0, 1]) for which lim_{k→∞} sup_{0≤t≤1} |x_{k}(t) − y_{k}(t)| = 0 then also the relation

lim_{k→∞} d(x_{k}(·), y_{k}(·)) = 0 holds with the metric d(·, ·) introduced at page 10 to metrize

the space D([0, 1]). Thus by Theorem A and formula (3.1) to prove Theorem 2 it is enough to show that the stochastic processes T_{k}(t), 0 ≤ t ≤ 1, converge weakly to the limit stochastic process described in Theorem 2.

The identity T_{k}(t) = T′_{k}(t) + T′′_{k}(t) holds, and the stochastic processes T′_{k}(t) and T′′_{k}(t), 1 ≤ t ≤ 1, appearing in this formula are independent. We shall show that the stochastic processes T′_{k}(·) converge to the Gaussian and the stochastic processes T′′_{k}(·) converge to the Poisson component of the limit process as k → ∞. To prove these statements first we have to clarify how the convergence of the canonical measures N_{k} on the strip R^{1} × [0, 1] to a canonical measure N_{0} is reflected in the behaviour of the canonical measures corresponding to the stochastic processes T′_{k}(·) and T′′_{k}(·).

Let us consider a sequence of positive numbers ε_{k} with which Lemma 1 holds. Let us define, by using the notation of Lemma 1, the measures N′_{k} on the interval [0, 1] concentrated in the points 0 ≤ u_{k,1} ≤ u_{k,2} ≤ ⋯ ≤ u_{k,n_{k}} = 1 introduced in the formulation of Theorem 2, for which N′_{k}(u_{k,l}) = G_{k,l}(ε_{k}) − G_{k,l}(−ε_{k}), k = 1, 2, …, where G_{k,l}( dx) = x^{2}F_{k,l}( dx), similarly to the formulation of Lemma 1. Let us also define the canonical measures N′′_{k} on the strip R^{1} × [0, 1] which are concentrated on the union of the lines R^{1} × u_{k,l}, 1 ≤ l ≤ n_{k}, and

N′′_{k}(B × {u_{k,l}}) = ∫_{B∩{u: |u|≥ε_{k}}} u^{2}F_{k,l}( du), 1 ≤ l ≤ n_{k}, k = 1, 2, ….

We claim that under the conditions of Theorem 2 the measures N′_{k} weakly converge to the measure N′_{0} defined by the relation N′_{0}(B) = N_{0}({0} × B) if B ⊂ [0, 1]. Beside this, the canonical measures N′′_{k} on the strip R^{1} × [0, 1] converge weakly to the canonical measure N′′_{0} defined by the relation N′′_{0}(B) = N_{0}(B \ ({0} × [0, 1])) if B ⊂ R^{1} × [0, 1].

To prove the above statements let us observe that if B ⊂ [0, 1] is a set with boundary ∂B such that λ(∂B) = 0 and C ⊂ R^{1} is a bounded set such that M_{0}(∂B) = 0, then under the conditions of Theorem 2 lim_{k→∞} N_{k}(C × B) = N_{0}(C × B). Furthermore,

lim_{k→∞} N′_{k}([0, 1]) = M({0}) by Lemma 1. First we show that for an arbitrary set B ⊂ [0, 1]

whose boundary satisfies the relation λ(∂B) = 0, lim sup_{k→∞} N′_{k}(B) ≤ N_{0}({0} × B) =

N′_{0}(B). Indeed, for all numbers δ > 0 there exists an interval [−η, η] with some η > 0 such that ±η is a point of continuity of the measure M and M_{0}([−η, η]) ≤ M_{0}({0}) + δ, and this implies that N_{0}([−η, η]×B) ≤ N_{0}({0}×B)+δ. Since ε_{k} → 0 if k → ∞, it follows from the above facts that lim sup_{k→∞} N′_{k}(B) ≤ lim_{k→∞} N_{k}(B × [−η, η]) = N_{0}([−η, η] × B) ≤

N_{0}({0} × B) + δ. As this relation holds for all numbers δ > 0, hence lim sup_{k→∞} N′_{k}(B) ≤

N′_{0}(B). By applying this inequality for both sets B and [0, 1] \ B we get that

N′_{0}([0, 1]) =M_{0}({0}) = lim_{k→∞} N′_{k}([0, 1]) ≤ lim sup_{k→∞} N′_{k}(B) + lim sup_{k→∞} N′_{k}([0, 1] \ B)
≤ N′_{0}(B) + N′_{0}([0, 1] \ B)) = N′_{0}([0, 1]).

This series of inequalities may be valid only if also the identity lim_{k→∞} N′_{k}(B) = N′_{0}(B) holds.

Let us also introduce the canonical measures N̄′_{k}, k = 1, 2, … and N̄′_{0} on the strip R^{1} × [0, 1] with the help of the formulas N̄′_{k}(A) = N_{k}(A ∩ [−ε_{k}, ε_{k}]), N̄′_{0}(A) = N_{0}(A ∩ {0} × [0, 1]), A ⊂ R^{1} ∩ [0, 1]. (In these formulas we lifted the measures N′_{k}, k = 1, 2, …, to the strip R^{1} × [0, 1].) Then the convergence of the measures N′_{k} to the measure N′_{0} implies the convergence of the canonical measures N̄′_{k} to the canonical measure N̄′_{0}. Furthermore, N′′_{k} = N_{k} − N̄′_{k}, k = 1, 2, …, and the convergence of the canonical measures N′′_{k} on the strip R^{1} × [0, 1] follows from the facts that the canonical measures N_{k} converge to the canonical measure N_{0} and the canonical measures converge to the canonical measure N̄′_{k}.

Let us define the random variables T′_{k,l} = ∑_{j=1}^{l} η′_{k,j}, 1 ≤ l ≤ n_{k}, T′_{k,0} = 0 and the

stochastic processes T′_{k}(t), T′_{k}(t) = T′_{k,l} if u_{k,l−1} ≤ t < u_{k,l}, T_{k,l}(1) = T′_{k,n_{k}}. Let us also consider the continuous function λ(t) = N_{0}({0} × [0, t]), 0 ≤ t ≤ 1 introduced in Part a) of Theorem 2. We claim that the stochastic processes T′_{k}(t), 0 ≤ t ≤ 1, weakly converge to the stochastic process W (λ(t)), 0 ≤ t ≤ 1, as k → ∞, where W (t), 0 ≤ t ≤ M({0}), is a standard Wiener process.

To prove the above statement let us introduce the numbers ū_{k,0} = 0, ū_{k,l} =

\frac{1}{U_{k}} ∑_{j=1}^{l} Var η′_{k,j}, 1 ≤ l ≤ n_{k}, k = 1, 2, …, where U_{k} = ∑_{j=1}^{n_{k}} Var η′_{k,j}, together with

the stochastic processes T̄′_{k}(·) which will be defined similarly to the stochastic processes T′_{k}(·) with the only difference that the numbers u_{k,l} are replaced by the numbers ū_{k,l} in the definition. Then we can state on the basis of the functional central limit theorem that the stochastic processes T̄′_{k}(t), 0 ≤ t ≤ 1, converge weakly to a standard Wiener process W (t) as k → ∞.

Let us define the monotone increasing and continuous functions λ_{k}(t), k = 1, 2, …, so that λ_{k}(u_{k,l}) = ū_{k,l}, 0 ≤ l ≤ k_{n}, and the function λ_{k}(·) is linear in the intervals [u_{k,l−1}, u_{k,l}], 1 ≤ k ≤ n_{k}. We shall show that

lim_{k→∞} U_{k} = M_{0}(\{0\}), lim_{k→∞} sup_{0≤t≤1} |λ_{k}(t) − \frac{λ(t)}{M_{0}(\{0\})}| = 0,  (3.2)

and this implies that the stochastic processes T′_{k}(t), 0 ≤ t ≤ 1 weakly converge to the stochastic process \sqrt{M_{0}(\{0\})}W (\frac{λ(t)}{M_{0}(\{0\})}) whose distribution agrees with the distribution of the stochastic process W (λ(t)), 0 ≤ t ≤ 1.

Indeed, Lemma 1 and formula (2.2) imply that

lim_{k→∞} U_{k} = lim_{k→∞} ∑_{j=1}^{n_{k}} Eξ′_{k,j}^{2} = M({0}).

We get similarly that lim_{k→∞} sup_{1≤l≤n_{k}} |ū_{k,l}U_{k} − ∑_{j=1}^{l} Eξ′_{k,j}^{2}| = 0, that is

lim_{k→∞} sup_{1≤l≤n_{k}} |ū_{k,l}M_{0}({0}) − N_{k}([−ε_{k}, ε_{k}]) × [0, u_{k,l}])| = 0.

The monotone functions h_{k}(t) = N_{k}([−ε_{k}, ε_{k}]) × [0, t]) converge to the monotone, continuous function λ(t) = N_{0}({0}×[0, t]) for all numbers 0 ≤ t ≤ 1, and because the above functions are monotone, and the limit function is continuous the above convergence is uniform. Hence lim_{k→∞} sup_{1≤l≤n_{k}} |ū_{k,l}M_{0}({0}) − λ(u_{k,l})| = 0. Thus

lim_{k→∞} sup_{1≤l≤n_{k}} |λ_{k}(u_{k,l}) − \frac{λ(u_{k,l})}{M_{0}(\{0\})}| = 0.

This implies relation (3.2).

To prove the weak convergence of the stochastic processes T′_{k}(t) = \frac{1}{\sqrt{U_{k}}} T̄′_{k} (λ_{k}(t)) to the stochastic process W (λ(t)) it is enough to show that

\sqrt{U_{k}}T̄′_{k} (λ_{k}(t)) − \sqrt{U_{k}}T̄′_{k} (\frac{λ(t)}{M_{0}(0)}) ⇒ 0,  (3.3)

where ⇒ denotes stochastic convergence. Indeed, Theorem A and relation (3.3) imply the desired statement, since the stochastic processes \sqrt{U_{k}}T̄′_{k} (\frac{λ(t)}{M_{0}(0)}) converge weakly to the stochastic process W (λ(t)). On the other hand, relation (3.3) follows from relation (3.2) and the result of the general theory by which the weak convergence of the stochastic processes \sqrt{U_{k}}T̄′_{k} (\frac{λ(t)}{M_{0}(0)}) to a stochastic process with continuous trajectories follows that the distributions of these processes are uniformly tight, i.e. for all numbers ε > 0 and η > 0 there exists a number δ = δ(ε, η) > 0 such that

P ( sup_{(s,t): |t−s|≤δ} |\sqrt{U_{k}}T̄′_{k} (\frac{λ(t)}{M_{0}(0)}) − \sqrt{U_{k}}T̄′_{k} (\frac{λ(s)}{M_{0}(0)})| > η) ≤ ε

for all indices k = 1, 2, . . .. (The number δ = δ(ε, η) does not depend on the index k.)

B.) The method of the proof. The study of the convergence of the Poissonian part.

In Part A of Section 3 we defined the representations T_{k}(t) = T′_{k}(t) + T′′_{k}(t) of the stochastic processes T_{k}(·), 1 ≤ t ≤ 1, k = 1, 2, …, introduced there and showed that Theorem 2 follows from the convergence of the distributions of the stochastic processes T_{k}(t), k = 1, 2, …, to the process S(t), defined in the formulation of Theorem 2, in the space D([0, 1]). Beside this, the stochastic processes T′_{k}(t) and T′′_{k}(t) are independent, and the stochastic processes T′_{k}(t) converge weakly to a Gaussian process with independent increments whose distribution can be described similarly to the limit process given in Theorem 2 with the difference that the measure N_{0}(·, ·) on the strip R^{1} × [0, 1]) is replaced by the measure N′_{0}(A) = N_{0}(A ∩ {0}) in formula (1.15), or more explicitly in the definition of the measure M_{0,u,v} appearing in this formula. Hence to complete the proof of Theorem 2 it is enough to show that the stochastic processes T′′_{k}(t) converge weakly to a stochastic process S′′_{0}(t), 0 ≤ t ≤ 1, with independent increments described by the canonical measure N′′_{0}(·, ·) on the strip R^{1} × [0, 1], given by the formula N′′_{0}(A) = N_{0}(A ∩ (R^{1} \ {0}) × [0, 1]). To simplify further discussions let us introduce the following definition.

The definition of the (time)–discretization of a stochastic process. Let Z(t), 0 ≤ t ≤ 1, be a stochastic process on the interval [0, 1], and let 0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{s} = 1, be a monotone sequence in the interval [0, 1]. Then the discretization of the stochastic process Z(t), 0 ≤ t ≤ 1, determined by the sequence of numbers 0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{s} = 1 is the stochastic process Z̄(t) = Z̄_{t_{0},t_{1},...,t_{s}}(t), 0 ≤ t ≤ 1, given by the formula

Z̄(t) = Z̄_{t_{0},t_{1},...,t_{s}}(t) = Z(t_{l−1}), if t_{l−1} ≤ t < t_{l}, 1 ≤ l ≤ s, Z̄(1) = Z(1).

Let us observe that the stochastic processes T′′_{k}(t) are discretizations (in the points 0 = u_{0,k} ≤ u_{1,k} ≤ ⋯ ≤ u_{k,n_{k}} = 1,) of infinitely divisible stochastic processes determined by Poisson processes with such counting measures ν_{k}(dx, dy) = \frac{N_{k}(dx,dy)}{x^{2}} for which the canonical measures N_{k} on the strip R^{1} × [0, 1] converge weakly to a canonical measure N′′_{0}, and the identity N_{k}({0} ∩ [0, 1]) = 0 holds. Hence we complete the proof of Theorem 2 if we prove the Statement formulated below. It can be considered as a generalization of Proposition 2 to random variables taking values in more general function spaces.

Statement. Let N_{k}, k = 0, 1, 2, …, be a sequence of canonical measures on the strip R^{1} ×[0, 1], and assume that these canonical measures N_{k} converge weakly to a canonical measure N_{0} if k → ∞, they satisfy condition b.) of Theorem 2, and N_{k}({0}×[0, 1]) = 0, k = 0, 1, 2, . . .. (This latter condition means that neither the processes determined by the measures N_{k}, k = 1, 2, …, nor the process determined by the limit measure N_{0} have a Gaussian component.) Let us define the canonical measures ν_{k}(dx, dy) = \frac{N_{k}(dx,dy)}{y^{2}}, k =

0, 1, 2, …, and consider Poisson fields X_{n}(k) = (X_{n}^{(1)}(k), X_{n}^{(2)}(k)), X_{n}(k) ∈ R^{1} × [0, 1], n = 1, 2, …, k = 1, 2, …, on the strip R^{1}×[0, 1] with canonical measures ν_{k}(dx, dy). Let us consider the infinitely divisible stochastic processes T_{k}(t), 0 ≤ t ≤ 1, k = 0, 1, 2, …, determined by these stochastic fields which can be considered as D([0, 1]) space valued random variables. The distributions of the stochastic processes T_{k}(t), k = 1, 2, …, converge to the distribution of the stochastic process T_{0}(t) in the space D([0, 1]) as k → ∞.

Beside this, let us have for all numbers k = 1, 2, … a partition 0 = u_{k,0} < u_{k,1} < ⋯ < u_{k,n_{k}} = 1 of the interval [0, 1], which satisfy the condition sup_{1≤j≤n_{k}} |u_{k,j} − u_{k,j−1}| =

0, and let us consider the discretizations T̄_{k}(t) = T̄_{k,u_{k,0},u_{k,1},...,u_{k,n_{k}}} (t) of the infinitely divisible processes T_{k}(t). The distributions of these discretizations T̄_{k}(t) of the stochastic processes T_{k}(t) also converge weakly to the distribution of the stochastic process T_{0}(t) in the space D([0, 1]).

The missing part of Theorem 2 agrees with the second part of the Statement about the convergence of the stochastic processes T̄_{k,u_{k,0},u_{k,1},...,u_{k,n_{k}}} (·) to the stochastic process T_{0}(·) if the same sequences of numbers 0 = u_{k,0} < u_{k,1} < ⋯ < u_{k,n_{k}} = 1 are considered as in Theorem 2.

Now we formulate two propositions make some comments about them and give the proof of the Statement with their help. These propositions will be proved in Part C.. Before their formulation let us recall how an infinitely divisible stochastic process (with nice trajectories in the space D([0, 1]) can be constructed by means of a Poissonian field with a counting measure which has some nice properties.

Let X_{n} = (X_{n}^{(1)}, X_{n}^{(2)}), X_{n} ∈ R^{1} × [0, 1], n = 1, 2, …, be a Poisson field with such a counting measure ν on the strip R^{1} × [0, 1] for which ν(R^{1} \ [−b, b] × [0, 1]) < ∞ and ∫_{(x,y): |x|≤b} x^{2}ν( dx, dy) < ∞ for all numbers b > 0. Then this Poisson field determines an infinitely divisible stochastic process T (t), 0 ≤ t ≤ 1, defined in the following way: Let us choose an appropriate sequence of numbers A_{L}, L = 1, 2, …, lim_{L→∞} A_{L} = 0, and put

T^{(L)}(t) =  ∑_{n: |X_{n}^{(1)}|>A_{L}, 0≤X_{n}^{(2)}≤t} X_{n}^{(1)}− E (  ∑_{n: |X_{n}^{(1)}|>A_{L}, 0≤X_{n}^{(2)}≤t}τ (X_{n}^{(1)})) , 0 ≤ t ≤ 1,

L = 1, 2, …, where the function τ(·) was defined in formula (1.2). It is proven in Lemma 2 in Section 5 of Part I that if the sequence A_{L} tends to zero sufficiently fast, then the limit T (t) = lim_{L→∞} T^{(L)}(t), 0 ≤ t ≤ 1, exists with probability 1, where the

limit is taken in supremum norm in the interval [0, 1]. Beside this, the trajectories of the so constructed stochastic process T (t) is an infinitely divisible stochastic process with cadlag trajectories. If N_{0} is a canonical measure on the strip R^{1} × [0, 1], then the distribution of the increments of the above defined stochastic process T (t), 0 ≤ t ≤ 1, determined by a Poisson field with counting measure ν_{0}(dx, dy) = \frac{N_{0}(dx,dy)}{x^{2}} is described by formula (1.15). We shall call the stochastic process T (t), 0 ≤ t ≤ 1, constructed in the above way the infinitely divisible process determined by the Poisson field X_{n} = (X_{n}^{(1)}, X_{n}^{(2)}), X_{n} ∈ R^{1} × [0, 1], n = 1, 2, . . ..

Proposition 3. Let a canonical measure N_{0} be given on the strip R^{1} × [0, 1] such that N_{0}({0} × [0, 1]) = 0. Let us also assume that the measure N_{0} satisfies Condition b.) of Theorem 2. Let us consider a Poisson field X_{n} = (X_{n}^{(1)}, X_{n}^{(2)}), X_{n} ∈ R^{1} × [0, 1], n = 1, 2, …, on the strip R^{1} × [0, 1]) with counting measure ν_{0}(dx, dy) = \frac{N_{0}(dx,dy)}{x^{2}}, and let T_{0}(t), 0 ≤ t ≤ 1, be the infinitely divisible process determined by this Poisson field. For all numbers ε > 0 and η > 0 there exists a number = δ(ε, η) such that for all sequences of number 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 for which the inequality sup_{1≤l≤s} |t_{l} − t_{l−1}| < δ

holds the stochastic process T_{0}(t) and its discretization T̄_{0}(t) = T̄_{0,t_{0},t_{1},...,t_{s}}(t) satisfies the inequality

P (d(T_{0}(·), T̄_{0}(·)) > η) < ε,  (3.4)

where d(·, ·) denotes the (simpler, not complete) metric introduced in the space D([0, 1]).

Let N_{k}, k = 1, 2, …, be canonical measures on the strip R^{1} ×[0, 1] which satisfy the relation N_{k}({0}×[0, 1]) = 0 and which converge to the canonical measure N_{0} considered in the previous paragraph. Let us define the measures ν_{k}(dx, dy) = \frac{N_{k}(dx,dy)}{x^{2}}, k = 1, 2, … and consider Poisson fields X_{n}(k) = (X_{n}^{(1)}(k), X_{n}^{(2)}(k)), X_{n}(k) ∈ R^{1} × [0, 1], n = 1, 2, …, k = 1, 2, …, on the strip R^{1} × [0, 1]) with counting measures ν_{k}(dx, dy). Let T_{k}(t), 0 ≤ t ≤ 1, k = 1, 2, …, denote the infinitely divisible stochastic processes determined by these Poissonian fields. Given some numbers ε > 0 and η > 0 there exists a number δ = δ(ε, η) and a threshold index k_{0} = k_{0}(η, ε) such that for all indices k ≥ k_{0} and sequences of numbers 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 for which the inequality sup_{1≤l≤s} |t_{l} −t_{l−1}| < δ holds the stochastic processes T_{k}(t) and their discretizations T̄_{k}(t) =

T̄_{k,t_{0},t_{1},...,t_{s}}(t) satisfy the inequality

P (d(T_{k}(·), T̄_{k}(·)) > η) < ε if k ≥ k_{0},  (3.5)

where d(·, ·) is the same metric in the space D([0, 1]) as that in formula (3.4).

Let us emphasize that the threshold index k_{0} = k_{0}(η, ε) in formula (3.5) can be chosen independently of the (sufficiently dense) sequence of numbers 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1.

Proposition 4. Let a sequence of canonical measures N_{k}, k = 1, 2, …, be given on the strip R^{1} × [0, 1] which converges to a canonical measure N_{0} as k → ∞, and N_{k}({0} × [0, 1]) = 0 for all numbers k = 0, 1, 2, . . .. Furthermore, let us fix a finite, monotone increasing sequence 0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{s} = 1 on the interval [0, 1]. Then for all indices k = 1, 2, … a Poisson field X_{n}(k) = (X_{n}^{(1)}(k), X_{n}^{(2)}(k)), X_{n}(k) ∈ R^{1}×[0, 1], n = 1, 2, …, can be constructed with canonical measures ν_{k}( dx, dy) = \frac{N_{k}( dx, dy)}{x^{2}} together with a Poisson field X′_{n}(k) = (X′_{n}^{(1)}(k), X′_{n}^{(2)}(k)), X′_{n}(k) ∈ R^{1} × [0, 1], n = 1, 2, …, with canonical measures ν_{0}( dx, dy) = \frac{N_{0}( dx, dy)}{x^{2}} in such a way that the infinitely divisible stochastic processes T_{k}(t) and T′_{k}(t), k = 1, 2, …, determined by these Poisson fields, or more explicitly their discretizations, the stochastic processes T̄_{k}(t) = T̄_{k,t_{0},t_{1},...,t_{s}}(t) and T̄′_{k}(t) = T̄′_{k,t_{0},t_{1},...,t_{s}}(t), 0 ≤ t ≤ 1, satisfy the relation

sup_{0≤t≤1} |T̄_{k}(t) − T̄′_{k}(t)| ⇒ 0 if k → ∞,  (3.6)

where ⇒ denotes stochastic convergence. (Let us remark that the distributions of the stochastic processes T′_{k}(·) and of their discretizations do not depend on the index k, since they are determined by a Poisson field with canonical measure ν_{0}.)

Let a sequence of canonical measures N_{k}, k = 1, 2, …, be given on the strip R^{1} × [0, 1] which satisfies the conditions of the Statement. Then with the help of Propositions 3 and 4 for all numbers ε > 0 and η > 0 a partition 0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{s} = 1 of the interval [0, 1] can be given together with two sequences of Poisson fields X_{n}(k) = (X_{n}^{(1)}(k), X_{n}^{(2)}(k)) and X′_{n}(k) = (X′_{n}^{(1)}(k), X′_{n}^{(2)}(k)), n = 1, 2, …, k = 1, 2, … on the strip R^{1} × [0, 1]) with counting measures ν_{k}(dx, dy) = \frac{N_{k}(dx,dy)}{x^{2}} and ν_{0}(dx, dy) = \frac{N_{0}(dx,dy)}{x^{2}} respectively which satisfy the following property. The infinitely divisible stochastic processes T_{k}(t) and T′_{k}(t), 0 ≤ t ≤ 1, determined by the Poisson fields X_{n}(k) and X′_{n}(k), n = 1, 2, …, k = 1, 2, …, and their discretizations, the stochastic processes T̄_{k}(·) = T̄_{k,t_{0},t_{1},...,t_{s}}(·) and T̄′_{k}(·) = T̄′_{k,t_{0},t_{1},...,t_{s}}(·), satisfy the relations

P (d(T′_{k}(·), T̄′_{k,t_{0},...,t_{s}}(·)) > η) < ε for all numbers k ≥ 1
P (d(T_{k}(·), T̄_{k,t_{0},...,t_{s}}(·)) > η) < ε, if k ≥ k_{0}
P ( sup_{0≤t≤1} |T̄_{k,t_{0},...,t_{s}}(t) − T̄′_{k,t_{0},...,t_{s}}(t)| > η) < ε, if k ≥ k_{0},

(3.7)

where k_{0} = k_{0}(ε, η) is an appropriate threshold index.

Indeed, by Proposition 3 the first two relations of formula (3.7) hold for an appropriate partition 0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{s} = 1 of the interval for all indices k ≥ k_{0} if k_{0} = k̄_{0}(ε, η) is sufficiently large. The validity of these relations does not depend on the way the Poisson fields X_{n}(k) and X′_{n}(k), n = 1, 2, …, k = 1, 2, …, with counting measures ν_{k} and ν_{0} are constructed. Then we can guarantee, because of Proposition 4, with an appropriate construction that also the third relation of formula (3.7) holds. (In this step we may increase the threshold index k_{0} if it is needed.)

Let us apply formula (3.7) with numbers ε_{j} = η_{j} = \frac{1}{j}. Then we can see that

there exists a monotone sequence of positive integers k_{0} (\frac{1}{j}) and for all j = 1, 2, … a

sequence of numbers 0 = t_{0}^{(j)} < t_{1}^{(j)} < t_{2}^{(j)} < ⋯ < t_{s_{j}}^{(j)} = 1 can be given which satisfy formula (3.7) for all j = 1, 2, … with the choice ε = η = \frac{1}{j} where the condition k ≥ k_{0}

is replaced by the condition k_{0} (\frac{1}{j}) ≤ k ≤ k_{0} (\frac{1}{j+1}), and we write 0 = t_{0}^{(j)} < t_{1}^{(j)} <

t_{2}^{(j)} < ⋯ < t_{s_{j}}^{(j)} = 1 instead of 0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{s} = 1, i.e. the partition of the interval [0, 1] we consider may depend on the index j. Hence under the conditions of the Statement two sequences of Poisson fields X_{n}(k) and X′_{n}(k), n = 1, 2, …, k = 1, 2, …, can be constructed on the strip R^{1} × [0, 1] with canonical measures ν_{k} and ν_{0} in such a way that the infinitely divisible stochastic processes T_{k}(t) and T′_{k}(t), k = 1, 2, …, 0 ≤ t ≤ 1, determined by them, together with their discretizations given by appropriate sequences of numbers 0 = t_{0}^{(k)} < t_{1}^{(k)} < t_{2}^{(k)} < ⋯ < t_{s_{k}}^{(k)} = 1, satisfy the relations

d (T′_{k}(·), T̄′_{k,t_{0}^{(k)},..., t_{s_{k}}^{(k)}}(·)) ⇒ 0 if k → ∞,  (3.8a)
d (T_{k}(·) , T̄_{k,t_{0}^{(k)},...,t_{s_{k}}^{(k)}}(·)) ⇒ 0 if k → ∞,  (3.8b)
sup_{0≤t≤1} |T̄_{k,t_{0}^{(k)},...,t_{s_{k}}^{(k)}}(t) − T̄′_{k,t_{0}^{(k)},...,t_{s}^{(k)}}(t)| ⇒ 0 if k → ∞.  (3.8c)

By Theorem A and relation (3.8a) the distributions of the stochastic processes T̄′_{k,t_{0}^{(k)},..., t_{s_{k}}^{(k)}}(·) converge to the distribution of the stochastic process T_{0}(·) in the space D([0, 1]). (Let us recall that the distributions of the stochastic processes T′_{k}(·) and T_{0}(·) agree.) Then by relation (3.8c) and Theorem A the distributions of the stochastic processes T̄_{k,t_{0}^{(k)},...,t_{s_{k}}^{(k)}}(·), and after this by relation (3.8b) and Theorem A the distributions of the stochastic processes T_{k}(·) converge to the distribution of the stochastic process T_{0}(·) in the space D([0, 1]). Thus we have proved the first part of the Statement. After this, the second part of the Statement follows from Theorem A and the second part of Proposition 3. Indeed, by this result

d (T_{k}(·) , T̄_{k,u_{k,0}^{(k)},...,u_{k,n_{k}}^{(k)}} (·)) ⇒ 0 if k → ∞.

(We exploit in this step of the proof that the threshold index k_{0} in formula (3.5) does not depend on the choice of the sufficiently dense partition 0 = t_{0} < t_{1} < · < t_{s} of the interval [0, 1].)

Let us remark that in Proposition 3 we have estimated the distance of a stochastic process and its discretization in the metric d(·, ·) introduced in the space D([0, 1]) and not in the supremum norm. We had to do so, since if the original stochastic processes have jumps, and we have to work with such stochastic processes, then these processes are far from their discretizations in the supremum norm. On the other hand, if the points of jumps are not too dense, which means that small intervals contain only at least one jump, then under general conditions a stochastic process and its sufficiently dense discretization are close to each other in the metric d(·, ·). In the next Lemma 2 we give an estimate for the d(·, ·) distance of two (simple) functions. It can help us to estimate the d(·, ·) distance of stochastic processes. In particular, it will be useful in the proof of Proposition 3.

Lemma 2. Let x(t) and y(t), 0 ≤ t ≤ 1, be two cadlag functions on the interval [0, 1] with p < ∞ numbers of jumps. (We assume that two functions have the same number of jumps.) Let us also assume that the values of the functions x(·) and y(·) agree after the j-th jump, 0 ≤ j ≤ p. Let there exist a finite monotone sequence of numbers 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 such that inf_{1≤l≤s} |t_{l} − t_{l−1}| ≤ δ with some number δ > 0,

and the function x(·) is constant in all intervals [t_{l−1}, t_{l}), 1 ≤ l ≤ s − 1. Furthermore, we assume that if the j-th point of jumps of the function x(·) is the point t_{l_{j}} with some l_{j} ≥ j, then the j-th jump of the other function y(·) is in a point of the interval (t_{l_{j−1}}, t_{l_{j}}], 1 ≤ j ≤ p. Then the inequality d(x(·), y(·)) < δ holds.

The proof of Lemma 2. Let u_{1}, …, u_{p} be the points of jumps of the function y(·). Let us define the following homeomorphism λ(·) of the interval [0, 1] onto itself: λ(u_{j}) = t_{l_{j}}, 1 ≤ j ≤ p, λ(0) = 0, λ(1) = 1, and let the function λ(·) be linear in the intervals [u_{j−1}, u_{j}], 1 ≤ j ≤ p, and [0, u_{1}], [u_{p}, 1]. (The number t_{l_{j}} is the j-th point of jump of the function x(·).) Then y(λ(·)) = x(·), and sup_{0≤t≤1} |λ(t) − t| ≤ δ. Hence d(x(·), y(·)) ≤ δ,

as we have claimed.

C.) The proof of Propositions 3 and 4.

The proof of Proposition 3. Let us choose a number α > 0 such that the numbers ±α are points of continuity of the canonical measure M_{0} on the real line, and M_{0}(−α, α]) < \frac{εη^{2}}{8} , where M_{0}(B) = N_{0}(B × [0, 1]), B ∈ R^{1}. Let us also introduce the canonical measures N′′_{0}(·) = N_{0,α}(·) and N′_{0,A_{L}}(·) = N′_{0,A_{L},α}(·) on the strip R^{1} × [0, 1] defined by the formulas N′′_{0}(B) = N_{0}(B ∩ {(x, y): |x| ≥ α}), N′_{0,A_{L}}(B) = N_{0}(B ∩ {(x, y): A_{L} ≤ |x| < α}) if B ∈ R^{1} ∩ [0, 1], and the numbers A_{L}, A_{L} > 0, are chosen in such a way that they can be applied in the regularized sums which define the stochastic process T_{0}(t) by means of a Poissonian field. Let us also introduce the measures ν′′_{0}( dx, dy) = \frac{N′′_{0}( dx, dy)}{x^{2}}

and ν′_{0,A_{L}}( dx, dy) = \frac{N′_{0,A_{L}} ( dx, dy)}{x^{2}} and consider a Poisson fields X′′_{n} = (X′′_{n}^{(1)}, X′′_{n}^{(2)}), n = 1, 2, …, with counting measure ν′′_{0} and the infinitely divisible field

T′′_{0}(t) = ∑_{n: X′′_{n}^{(1)}>α, 0≤X′′_{n}^{(2)}≤t} X′′_{n}^{(1)}−E (  ∑_{n: X′′_{n}^{(1)}>α, 0≤X′′_{n}^{(2)}≤t} τ (X′′_{n}^{(1)})) , 0 ≤ t ≤ 1,

determined by it. Let us consider similarly a Poisson field X′_{n,A_{L}} = (X′_{n,A_{L}}^{(1)}, X′_{n,A_{L}}^{(2)}), n = 1, 2, … with counting measure ν′_{0,A_{L}} and the infinitely divisible stochastic process

T′_{0,A_{L}}(t) = ∑_{n: A_{L}≤X′_{n}^{(1)}<α, 0≤X′_{n}^{(2)}≤t} X′_{n}^{(1)} − E (  ∑_{n: A_{L}≤X′_{n}^{(1)}<α, 0≤X′_{n}^{(2)}≤t} τ (X′_{n}^{(1)})) ,

0 ≤ t ≤ 1, determined by this Poisson field. Put T_{0,A_{L}}(t) = T′_{0,A_{L}}(t) + T′′_{0}(t), 0 ≤ t ≤ 1. The stochastic processes T_{0,A_{L}}(·) converge with probability 1 to the stochastic process T_{0}(·) in the supremum norm. Hence to prove formula (3.4) it is enough to show that

P (d(T_{0,A_{L}}(·), T̄_{0,A_{L},t_{0},··· ,t_{s}}(·)) > η) < ε for all numbers L ≥ L_{0},  (3.9)

where L_{0} is an appropriate number, and T̄_{0,A_{L},t_{0},·,t_{s}}(·)) denotes the discretization of the stochastic process T_{0,A_{L}}(·). Indeed,

{ω: d(T_{0}, (·, ω), T̄_{0}(·, ω)) > η} ⊂ lim inf_{L→∞} {ω: d(T_{0,A_{L}}(·, ω), T̄_{0,A_{L},t_{0},··· ,t_{s}}(·, ω)) > η} ,

hence formula (3.9) implies formula (3.4).

We can write

P (d(T_{0,A_{L}}(·), T̄_{0,A_{L},t_{1},...,t_{s}}(·)) > η) ≤ P ( sup_{0≤t≤1} |T′_{0,A_{L}}(t)| > \frac{η}{2} )

+ P (d(T′′_{0}(·), T̄′′_{0,t_{1},...,t_{s}}(·)) > \frac{η}{2} ) ,

hence to prove formula (3.9) it is enough to show that

P ( sup_{0≤t≤1} |T′_{0,A_{L}}(t)| > \frac{η}{2} ) < \frac{ε}{2}, if L ≥ L_{0},  (3.10)

and

P (d(T′′_{0}(·), T̄′′_{0,t_{1},...,t_{s}}(·)) > \frac{η}{2} ) < \frac{ε}{2}, if sup_{1≤l≤s} |t_{l} − t_{l−1}| < δ  (3.11)

with some appropriate number δ > 0.

As T′_{0,A_{L}}(t) is a stochastic process with independent increments, and its trajectories are cadlag functions hence we can write with the help of the Kolmogorov inequality that

P ( sup_{0≤t≤1} |T′_{0,A_{L}}(t)| > \frac{η}{2} ) ≤ \frac{4ET′_{0,A_{L}}(1)^{2}}{η^{2}} = \frac{4}{η^{2}} ∫ u^{2}μ′_{0,A_{L}}( du) ≤ \frac{4}{η^{2}} M_{0}([−α, α]) < ε,

where the measure μ′_{0,A_{L}} is defined by the relation μ′_{0,A_{L}}(B) = ν′_{0,A_{L}}(B × [0, 1]) for all measurable sets B ⊂ R^{1}. Hence inequality (3.10) holds.

To prove formula (3.11) let us first introduce the function λ_{0}(t) = λ_{0,α}(t) defined by the formula λ_{0}(t) = ν′′_{0}(R^{1} × [0, t]) = ∫_{\{(x,y): |x|>α, 0≤|y|<t\}} \frac{N_{0}( dx, dy)}{x^{2}}, 0 ≤ t ≤ 1. Let us remark that because of Condition b.) of Theorem 2 the function λ_{0}(·) is continuous in the interval [0, 1]. We claim that thee exists some number δ > 0 such that for all partitions 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 of the interval [0, 1]

∑_{l=1}^{s} (λ_{0}(t_{l}) − λ_{0}(t_{l−1}))^{2} < ε if |t_{l} − t_{l−1}| < δ for all numbers 1 ≤ l ≤ s.  (3.12)

Indeed, as the function λ_{0}(·) is uniformly continuous, hence there exists a number δ > 0 such that |λ_{0}(t) − λ_{0}(s)| < \frac{ε}{λ(1)} if |t − s| < δ. Furthermore, the function λ_{0}(·) is mono-

tone increasing. Hence ∑_{l=1}^{s} (λ_{0}(t_{l}) − λ_{0}(t_{l−1}))^{2} ≤ sup_{1≤l≤s} |λ_{0}(t_{l}) − λ_{0}(t_{l−1})| ∑_{l=1}^{s} |λ_{0}(t_{l}) −

λ_{0}(t_{l−1})| < ε if |t_{l} − t_{l−1}| < δ for all numbers 1 ≤ l ≤ s, i.e. formula (3.12) is valid.

A Poisson distributed random variable with parameter λ takes a value more than or equal to two with probability 1 − e^{−λ} − λe^{−λ} ≤ \frac{λ^{2}}{2} . Hence the probability of the event that a Poisson field X′′_{n} = (X′′_{n}^{(1)}, X′′_{n}^{(2)}), n = 1, 2, …, on the strip R^{1} × [0, 1] with counting measure ν′′_{0} contains at least two such points X′′_{n_{1}} and X′′_{n_{2}} whose second coordinates X′′_{n_{1}}^{(2)} and X′′_{n_{2}}^{(2)} are in an interval [s, t], 0 ≤ s < t ≤ 1, is less than \frac{1}{2}ν′′(R^{1} × [s, t])^{2} = \frac{1}{2}(λ(t) − λ(s))^{2}. Hence by formula (3.12) for a Poisson field X′′_{n} = (X′′_{n}^{(1)}, X′′_{n}^{(2)}), n = 1, 2, …, with counting measure ν′′_{0} and a partition 0 ≤ t_{0} < t_{1} < ⋯ < t_{s} = 1 of the interval [0, 1] the probability of the event

A(t_{1}, …, t_{s}) = {ω: #{n: t_{l−1} ≤ X′′_{n}^{(2)}(ω) ≤ t_{l}} ≤ 1 for all numbers 1 ≤ l ≤ s}

can be estimated as

1 − P (A(t_{1}, …, t_{s})) = P (Ω \ A(t_{1}, …, t_{s})) ≤ \frac{1}{2} ∑_{l=1}^{s} (λ_{0}(t_{l}) − λ_{0}(t_{l−1}))^{2} ≤ \frac{ε}{2} (3.13)

if sup_{1≤l≤s} |t_{l} − t_{l−1}| < δ with a sufficiently small δ > 0.

Let us fix a number δ > 0 with which formula (3.13) holds together with the inequality δ < \frac{η}{2} . Then formula (3.13) means that on a set of probability greater than 1 − \frac{ε}{2} the cadlag functions x(t) = T̄′′_{0,t_{0},t_{1},...,t_{s}}(t, ω) and y(t) = T′′_{0}(t, ω) together with the partition 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 of the interval [0, 1] satisfy the conditions of Lemma 2, and d(T′′_{0}(·, ω), T̄′′_{0,t_{0},··· ,t_{s}}(·, ω)) ≤ δ < \frac{η}{2} on a set of probability greater than 1 − \frac{ε}{2}. This fact implies relation (3.11) and as a consequence also relation (3.4).

To prove formula (3.5) it is enough to prove the following analogs of formulas (3.10) and (3.11).

P ( sup_{0≤t≤1} |T′_{k,A_{L}}(t)| > \frac{η}{2} ) < \frac{ε}{2}, if k ≥ k_{0} and L ≥ L_{0}  (3.14)

with an appropriate threshold index k_{0} = k_{0}(ε, η), and

P (d(T′′_{k}(·), T̄′′_{k,t_{1},...,t_{s}}(·)) > \frac{η}{2} ) < \frac{ε}{2}, if k ≥ k_{0}, and sup_{1≤l≤s} |t_{l} − t_{l−1}| < δ (3.15)

with an appropriate threshold index k_{0} = k_{0}(ε, η) and number δ > 0.

Formula (3.14) can be proved similarly to formula (3.10). The only difference is that now we exploit that since the numbers ±α are points of continuity of the measure M_{0}, and M_{0}([−α, α]) < \frac{εη^{2}}{8} hence also the relation M_{k}(−α, α) < \frac{εη^{2}}{8} holds if k ≥ k_{0} with an appropriate threshold index k_{0}. (We define the measures M_{k}, k = 1, 2, …, on the real line, analogously to the definition of the measure M_{0}, by the formula M_{k}(B) = N_{k}(B × [0, 1]) for all measurable sets B ∈ R^{1}.)

Formula (3.15) can be proved similarly to formula (3.11). The only difference is that now we have to prove and apply the following analog of relation (3.12).

Let us introduce the functions λ_{k}(t) = λ_{k,α}(t) defined by the formula λ_{k}(t) = ν′′_{k}(R^{1} × [0, t]) = ∫_{\{(x,y): |x|>α, 0≤|y|<t\}} \frac{N_{k}( dx, dy)}{x^{2}}, 0 ≤ t ≤ 1 for all indices k = 1, 2, . . .. Then there exists number δ > 0 and threshold index k_{0} = k_{0}(δ) in such a way that for all partitions 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 of the interval [0.1]

∑_{l=1}^{s} (λ_{k}(t_{l}) − λ_{k}(t_{l−1}))^{2} ≤ ε if k ≥ k_{0} and |t_{l} − t_{l−1}| < δ for all numbers 1 ≤ l ≤ s.

(3.16)

Let us emphasize the threshold index k_{0} in formula (3.16) depends only on the number δ > 0 and not on the partition 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 of the interval [0, 1].

Let us prove formula (3.16) first in the special case if the partition 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 of the interval [0, 1] satisfies not only the inequality |t_{l} − t_{l−1}| < δ, but also the inequality |t_{l} − t_{l−1}| ≥ \frac{δ}{2} for all numbers 1 ≤ l ≤ s. The convergence of the canonical measures N_{k} to the canonical measures N_{0} and the continuity of the measure λ_{0}(·) imply that the monotone functions λ_{k}(·) converge in all points of the interval [0, 1] to the monotone and continuous function λ_{0}(·). The properties of the functions λ_{k}(·) also imply that the convergence lim_{k→∞} λ_{k}(t) = λ_{0}(t), 0 ≤ t ≤ 1, is uniform. Beside

this, the sum at the right-hand side of formula (3.16) contains at most \frac{2}{δ} terms. Hence formula (3.12) implies formula (3.16) with the same number δ and a sufficiently large threshold index k_{0}(δ).

Let us now consider a partition 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 of the interval such that sup_{1≤s≤k} |t_{l} − t_{l−1}| < \frac{δ}{2} where formula (3.16) holds in the special case considered in

the previous paragraph with the number δ. It is not difficult to see that the sequence of numbers 0 = t_{0} < t_{1} < ⋯ < t_{s} = 1 has a subsequence 0 = t_{j_{0}} < t_{j_{1}} < ⋯ < t_{j_{p}} = 1 such that \frac{δ}{2} ≤ |t_{j_{u}} − t_{j_{u−1}} | < δ for all numbers 1 ≤ u ≤ p. Let us take such a subsequence. Then we can write with the help of the already proven case that

∑_{l=1}^{s} (λ_{k}(t_{l}) − λ_{k}(t_{l−1}))^{2} ≤ ∑_{u=1}^{p} (λ_{k}(t_{j_{u}}) − λ_{k}(t_{j_{u−1}}))^{2} < ε

if k ≥ k_{0}(δ). In such a way we have proved formula (3.16) (with the choice \frac{δ}{2} instead of the number δ.) After the proof of formulas (3.15) and (3.16) formula (3.5) can be proved in the same way as formula (3.4).

The proof of Proposition 4. The proof of Proposition 4 is based on the following observation. To determine the difference T̄_{k}(·)−T̄′_{k}(·) of the discretizations of the processes T_{k}(·) and T′_{k}(·) we need not know the precise values of the Poisson fields X_{k,n} = (X_{k,n}^{(1)}, X_{k,n}^{(2)})

and X′_{k,n} = (X′_{k,n}^{(1)}, X′_{k,n}^{(2)}), n = 1, 2, …, k = 1, 2, …, which determine these processes. The knowledge of the values of the second coordinates of these fields is not necessary, it is enough to know in which one of the intervals [t_{l−1}, t_{l}] they lie. Hence in the first step of the construction we do not decide the precise value of the Poisson fields we have to define. In such a way in the first step of the construction a coupling problem has to be handled which can be solved relatively simply with the help of Proposition 2. Then the construction can be completed by an appropriate randomization.

To work out the details first we introduce some notations. Let us define the canonical measures Ñ_{k,l}, k = 0, 1, 2, …, 1 ≤ l ≤ s, on the real line by the formula Ñ_{k,l}(B) = N_{k}(B × (t_{l−1}, t_{l}]), k = 0, 1, 2, …, 1 ≤ l ≤ s. Let us then define the canonical measures Ñ′_{k,l} on the strip R^{1}×[0, 1] by the formulas Ñ′_{k,l}(B×{t_{l}}) = Ñ′_{k,l}(B), B ⊂ R^{1}, k = 0, 1, 2, …, 1 ≤ l ≤ s, i.e. let the measure Ñ′_{k,l} be the shift of the measure Ñ_{k,l} from the real line to the line {(x, y): y = t_{l}} parallel to the coordinate axis in R^{2}. Let us also define the canonical measures Ñ_{k}, k = 0, 1, 2, …, on the strip R^{1} × [0, 1] as

Ñ_{k} = ∑_{l=1}^{s} Ñ′_{k,l}, k = 0, 1, 2, . . ..

Let us define the measures ν̃_{k,l}(dx) = \frac{Ñ_{k,l}(dx)}{x^{2}}, ν̃′_{k,l}(dx) = \frac{Ñ′_{k,l}(dx)}{x^{2}} and ν̃_{k}(dx, dy) =

\frac{Ñ_{k}(dx,dy)}{x^{2}}, k = 0, 1, 2, …, l = 1, 2, …, s. By Proposition 2 some Poisson fields ξ_{k,l}(n), n = 1, 2, …, k = 1, 2, …, l ≤ 1 ≤ s, can be constructed with counting measures ν̃_{k,l} together with some other Poisson fields ξ′_{k,l}(n), n = 1, 2, …, k = 1, 2, …, 1 ≤ l ≤ s, with counting measures ν̃_{0,l} in such a way that the random variables U_{k,l} and U′_{k,l} with infinitely divisible distributions determined by these Poisson fields satisfy the relations U′_{k,l} − U_{k,l} ⇒ 0 as k → ∞ for all numbers 1 ≤ l ≤ s, where ⇒ denotes stochastic

convergence. Let us also define the random variables T̃_{k,j} = ∑_{l=1}^{j} U_{k,l}, T̃′_{k,j} = ∑_{l=1}^{j} U′_{k,l},

k = 1, 2, …, 1 ≤ j ≤ s. Then also the relation sup_{1≤j≤s} |T̃_{k,j} − T̃′_{k,j}| ⇒ 0 holds as k → ∞.

Now we begin the construction of Poisson fields for which the discretizations of the stochastic processes T_{k}(t) and T′_{k}(t) determined by them satisfy formula (3.6). Let us consider the measures ν_{k,l}, k = 0, 1, 2, …, 1 ≤ l ≤ s, which are restrictions of the measures ν_{k}(dx, dy) = \frac{N_{k}(dx,dy)}{x^{2}} to the strip R^{1} × (t_{l−1}, t_{l}]. Then for all integers k = 0, 1, 2 …, 1 ≤ l ≤ s, points x ∈ R^{1}, and measurable sets B ∈ R^{1} × (t_{l−1}, t_{l}] there exist such “conditional measures” ν_{k,l}(B|x) on the interval (t_{l−1}, t_{l}] for which ν_{k,l}( · |x) is a probability measure in the strip R^{1} × (t_{l−1}, t_{l}] for all numbers x ∈ R^{1}, ν_{k,l}(B| ·) is a measurable function on the real line for all measurable sets B ⊂ R^{1} × (t_{l−1}, t_{l}], and

ν_{k,l}(B) = ∫ ν_{k,l}(B|x)ν̃′_{k,l}( dx) for all measurable sets B ⊂ R^{1} × (t_{l−1}, t_{l}] (3.17)

for all numbers k = 0, 1, 2, … and 1 ≤ l ≤ s, where ν̃_{k,l} is the measure defined in the previous paragraph. The existence of a “conditional measure” ν_{k,l}( · | ·) satisfying relation (3.17) is a consequence of a classical result of probability theory about the existence of regular conditional distributions.

Now we construct Poisson fields X_{k,n} = (X_{k,n}^{(1)}, X_{k,n}^{(2)}), n = 1, 2, …, k = 1, 2, …

with counting measures ν_{k} and Poisson fields X′_{k,n} = (X′_{k,n}^{(1)}, X′_{k,n}^{(2)}), n = 1, 2, …, k = 1, 2, … with counting measures ν_{0} such that the stochastic processes T_{k}(t) and T′_{k}(t) determined by them satisfy Proposition 4. Let us consider the already constructed Poisson fields ξ_{k,l}(n), n = 1, 2, …, k = 1, 2, …, l ≤ 1 ≤ s, with counting measures ν̃_{k,l} and the Poisson fields ξ′_{k,l}(n), n = 1, 2, …, k = 1, 2, …, 1 ≤ l ≤ s, with counting measures ν̃_{0,l}. For all points ξ_{k,l}(n) and ξ′_{k,l}(n) of these Poisson fields we shall construct random variables η_{k,l}(n) and η′_{k,l}(n) in a random way, and the Poisson fields we want to construct will consist of the points (ξ_{k,l}(n), η_{k,l}(n)) and (ξ′_{k,l}(n), η′_{k,l}(n)). Let us construct for all random variables ξ_{k,l}(n) a ν_{k,l}( · |ξ_{k,l}(n)) distributed random variable η_{k,l}(n) and for all random variables ξ′_{k,l}(n) a ν_{0.l}( · |ξ′_{k,l}(n)) distributed random variables η′_{k,l}(n) on the interval [t_{l−1}, t_{l}]. Let us construct these random numbers independently

of each other. For a fixed index k let the Poison field X_{k,n} = (X_{k,n}^{(1)}, X_{k,n}^{(2)}), n = 1, 2, …, k = 1, 2, … with counting measure ν_{k} consist of the previously constructed pairs of points (ξ_{k,l}(n), η_{k,l}(n)), n = 1, 2, …, and similarly let the Poisson field X′_{k,n} =

(X′_{k,n}^{(1)}, X′_{k,n}^{(2)}), n = 1, 2, …, k = 1, 2, … with counting measure ν_{0} consist of the pairs of points (ξ′_{k,l}(n), η′_{k,l}(n)), n = 1, 2, …, 1 ≤ l ≤ s. We claim that the above constructed Poisson fields satisfy Proposition 4.

We shall prove that if the rectangle B × [t_{l−1}, t_{l}] satisfies the property ν_{k,l}(B × [t_{l−1}, t_{l}]) < ∞, then the points falling to this rectangle B × [t_{l−1}, t_{l}] define a Poisson field with counting measure ν_{k,l} on this rectangle. We shall show this for all numbers k = 1, 2, … and 1 ≤ l ≤ s. Beside this we claim that if the property ν_{0,l}(B × [t_{l−1}, t_{l}]) < ∞ holds, then the points (ξ′_{k,l}(n), η′_{k,l}(n)), n = 1, 2, …, falling to the rectangle B ×[t_{l−1}, t_{l}] define a Poisson field with counting measure ν_{0,l} on this rectangle for all numbers k = 1, 2, … and 1 ≤ l ≤ s. These facts imply that we have really constructed Poisson fields with the right counting measure. These statements can be simply proved with the help of the following observation. The distributions of these point processes agree with the distributions of the point process we get in the following way: Let us choose randomly many number of points with Poisson distribution with parameter ν_{k,l}(B × [t_{l−1}, t_{l}]) and drop them randomly to the rectangle B × [t_{l−1}, t_{l}] independently of each other with distribution \frac{μ_{k,l}(dx,dy)}{ν_{k,l}(B×[t_{l−1},t_{l}])}. Such constructions supply Poisson fields with the right counting measure.

Finally we remark that the above constructed Poisson fields are such that the discretizations of the infinitely divisible stochastic processes T_{k}(·) and T′_{k}(·) determined by them satisfy the inequalities T̄_{k,t_{0},··· ,t_{s}}(t) = T̃_{k,j−1} and T̄′_{k,t_{0},··· ,t_{s}}(t) = T̃′_{k,j−1} if t_{j−1} < t ≤ t_{j}, 1 ≤ j ≤ s, and T̄_{k,t_{0},··· ,t_{s}}(1) = T̃_{k,s}, T̄′_{k,t_{0},··· ,t_{s}}(1) = T̃′_{k,s}, k = 1, 2, …, Hence the processes T̄_{k,t_{0},··· ,t_{s}}(·) and T̄′_{k,t_{0},··· ,t_{s}}(·) satisfy formula (3.7). Proposition 4 is proved.

Appendix. The proof of Theorem A.

The proof of Theorem A. It is enough to prove the statement formulated in general separable metric spaces. The weak convergence of the random variables S_{k} to a probability measure μ can be formulated so that the distributions μ_{k} of the random variables S_{k} satisfy the relation lim sup_{k→∞} μ_{k}(F) ≤ μ(F) for all closed sets F ⊂ X. We shall show that

under the conditions of Theorem A the distributions μ̄_{k} of the random variables T_{k} also satisfy the relation lim sup_{k→∞} μ̄_{k}(F) ≤ μ(F) for all closed sets F ⊂ X. (Let us remark

that the characterization of the weak convergence applied in this proof is valid in all separable metric spaces. We do not have to assume that the metric space is complete. (See for instance Theorem 2.1 in the book of P. Billingsley Convergence of Probability Measures.)

Let us fix some number ε > 0. As F = ⋂_{n=1}^{∞} F_{\frac{1}{n}}, where F_{a} = \{x: ρ(x, F) ≤ a\},

hence there exists a number δ = δ(ε) > 0 such that μ(F) ≥ μ(F_{δ}) − ε. Furthermore, the inequality μ_{k}(F_{δ}) < μ(F_{δ}) + ε holds if k ≥ k_{0} = k_{0}(ε, δ, F). As ρ(S_{k}, T_{k}) tends to zero stochastically, hence also the inequality μ̄_{k}(F) = P (T_{k} ∈ F) ≤ P (S_{k} ∈ F_{δ}) + P (ρ(S_{k}, T_{k}) > δ) ≤ μ_{k}(F_{δ}) + ε, holds if k ≥ k_{0} and the threshold index k_{0} = k_{0}(ε, δ, F) is chosen sufficiently large. The above inequalities imply that μ̄_{k}(F) ≤ μ_{k}(F_{δ}) + ε ≤ μ(F_{δ}) + 2ε ≤ μ(F) + 3ε if k ≥ k_{0}(ε, δ, F). The above inequality holds for all numbers ε > 0, and this implies that lim sup_{k→∞} μ̄_{k}(F) ≤ μ(F). Thus Theorem A is proved.