ASYMPTOTIC DISTRIBUTIONS

FOR WEIGHTED U-STATISTICS

Péter Major

Mathematical Institute of the Hungarian Academy of Sciences

Abstract. We prove limit theorems for weighted U-statistics and express the limit by means of multiple stochastic integrals. This is a generalization of the paper of K. A. O’Neil and R. A. Redner [9]. In that paper the method of moments was applied which does not work in the general case. Hence we had to work out a diffferent method. In particular, in Theorem 4 we describe the limit of a model proposed by O’Neil and Redner. In this model the weight functions cause an intricate cancelletion, and the limit can be presented as a sum of multiple stochastic integrals with different multiplicities.

1. Introduction. In this paper we investigate the limit behavior of weighted U-statistics which means statistics of the following form:

U_{n} = ∑_{1≤j_{1}<j_{2}···<j_{k}≤n} a(j_{1}, …, j_{k})f(X_{j_{1}}, …, X_{j_{k}}) .  (1.1)

Here X_{1}, …, X_{n} are iid. random variables with uniform distribution in the interval [0, 1], the functions a(x_{1}, …, x_{k}) and f(x_{1}, …, x_{k}) are symmetric, i.e. they are invariant under all permutations of their arguments, and the function f also satisfies the condition

∫_{[0,1]^{k}} f^{2}(x_{1}, …, x_{k}) dx_{1} … dx_{k} < ∞ .  (1.2)

The expression (1.1) is a generalization of the usual (unweighted) U-statistics, investigated e.g. in [1], because of the appeareance of a weight function a(x_{1}, …, x_{k}) in it. The assumption that the sequence of iid. random variables X_{1}, …, X_{n} is uniformly distributed is not a real restriction. If its distribution function is F (x), then the sequence F (X_{1}), …, F (X_{n}) is uniformly distributed, and the statistics U_{n} do not change if the function f(x_{1}, …, x_{k}) is replaced by f(F^{−1}(x_{1}), …, F^{−1}(x_{k})) and the random variables X_{j}, by F (X_{j}), j = 1, …, n. In most results of this paper we restrict our attention to the so-called degenerate U-statistics, i.e. we assume that

∫ f(y, x_{2}, …, x_{k}) dy = 0 for all x_{2}, …, x_{k} .  (1.3)

The investigation of U-statistics with general kernel functions f can be reduced to this special case by means of the Hoeffding decomposition. (See e.g. Appendix A in [1].) This gives the following representation of a symmetric function

Typeset by AMS-TEX

f(x_{1}, …, x_{k}) with k arguments: There exists a (unique) sequence of symmetric functions f_{s} = f_{s}(x_{1}, …, x_{s}), s = 1, …, k, and a constant f_{0} such that

f(x_{1}, …, x_{k}) = f_{0} + ∑_{s=1}^{k} ∑_{{i_{1},...,i_{s}}⊂{1,...,k}} f_{s}(x_{i_{1}}, …, x_{i_{s}}) ,  (1.4)

and the functions f_{s} are degenerate, i.e.

∫ f_{s}(y, x_{2}, …, x_{s}) dy = 0 for all x_{2}, …, x_{s}, 2 ≤ s ≤ k .  (1.4′)

Because of this decomposition the investigation of the limit behavior of the statistics U_{n} for n → ∞, defined in (1.1), with a general kernel functions f can be reduced to that with degenerate kernel functions. In the case of unweighted U-statistics when a(x_{1}, …, x_{k}) ≡ 1 the first non-vanishing term in (1.4), i.e. the function f_{s} with the smallest index s in the Hoeffding representation such that f_{s} is not identically zero, gives the dominating contribution to the U-statistics. For typical weighted U-statistics the case is similar, but the general situation is more complex. In this respect we refer to the second Section of [9] and return to this question in Section 2.

Our investigation was motivated by a recent paper of Kevin A. O’Neil and Richard A. Redner [9] where asymptotic distribution of weighted U-statistics of degree two was investigated. This means the investigation of statistics defined by formula (1.1) in the special case k = 2. The authors of this paper proved the existence of a limit distribution with an appropriate normalization by showing the convergence of the moments. This method works only if the limit distribution is determined by its moments. This property holds for U-statistics of degree one or two. But if k ≥ 3, then for U-statistics defined by formula (1.1) (with a degenerate kernel f satisfying relation (1.3)) such a limit distribution appears which is not determined by its moments. Hence in this case a different method has to be applied. The aim of the present paper is to find such a method and to give an explicit expression for the appearing limit. Let us first explain why the limit distribution of U-statistics is not determined by its moments for k ≥ 3.

The limit of unweighted (degenerate) U-statistics, with normalization n^{−k/2}, can be expressed by means of k-fold Wiener–Itô integrals with respect to a Wiener process. On the other hand, the following result is known about the the tail behavior of multiple stochastic integrals. (See e.g. [8] or Section 6 in [7].) If I_{k} = ∫ f(x_{1}, …, x_{k}) B( dx_{1}) … B( dx_{k}) is a k-fold Wiener-Itô integral with respect to a Gaussian random measure, then

C_{1} exp {−L_{1}x^{2/k}} < P (|I_{k}| > x) < C_{2} exp {−L_{2}x^{2/k}} for all x > 1

with some appropriate constants C_{1} > 0, C_{2} > 0 and L_{1} > L_{2} > 0. For us the left-hand side of the last inequality is interesting. If a distribution function F (x) decreases at plus and minus infinity exponentially fast, then its moments determine its distribution. On the other hand, if F (−x) + 1 − F (x) > C exp{−Lx^{α}} with some 0 ≤ α < 1 and C > 0, L > 0 then we cannot say that F is determined by its moments. (See e.g [2] for an example.) This second case appears in the case of k-fold stochastic integrals with k ≥ 3.

The case of weighted U-statistics is similar. The only difference is that for typical weight functions a(j_{1}, …, j_{k}) the limit of the statistics can be expressed by a k-fold stochastic integral with respect to a Wiener sheet instead of a Wiener process. The Wiener sheet is the natural two-dimensional analogue of a Wiener process. It is a two-dimensional Gaussian process B(x, y), 0 ≤ x, y ≤ 1, with expectation zero whose increments B(x_{2}, y_{2})+B(x_{1}, y_{1})−B(x_{1}, y_{2})−B(x_{2}, y_{1}) on disjoint rectangles [x_{1}, x_{2}]×[y_{1}, y_{2}] are independent with variance (x_{2}−x_{1})(y_{2}−y_{1}).

Let us briefly explain our approach. First we give a short explanation about how to handle unweighted U-statistics and try to adapt it to the case of weighted U-statistics. Let F_{n}(x) denote the empirical distribution function determined by the sample X_{1}, …,X_{n}. In the case of unweighted U-statistics when the weightfunction a(·) is identically one formula (1.1) can be rewritten as

U_{n} = \frac{n^{k}}{k!} ∫ ′ f(x_{1}, …, x_{k}) F_{n}( dx_{1}) … F_{n}( dx_{k}) ,  (1.5)

where ∫′ denotes that the hyperplanes x_{i} = x_{j} for i ≠ j are cut out from the domain of integration. Since we consider the degenerate case when relation (1.3) holds, the expression (1.5) does not change if F_{n}(x) is replaced by F_{n}(x) − x. We recall that \sqrt{n} (F_{n}(x) − x) ⇒ B_{0}(x), where B_{0}(x) is a Brownian bridge. Hence, it is natural to expect that we commit a small error by replacing \sqrt{n} (F_{n}(x) − x) by B_{0}(x) or, by exploiting formula (1.3) again, by a Wiener process B_{0}(x) + xξ, where ξ is a standard normal random variable independent of the Brownian bridge B_{0}(x). The last step is useful, because the theory of multiple stochastic integral is applicable with respect to Gaussian processes with independent increments like the Wiener process, but not with respect to a Brownian bridge. The above argument supplies an informal proof of the limit theorem for the distribution of unweighted U-statistics, and a rigorous proof can be obtained by justifying the above manipulations.

If we want to adapt the above argument to weighted U-statistics we meet some problems at the start. Formula (1.5) does not hold any longer, moreover U_{n} cannot be expressed as a functional of F_{n}(x), since it is not a function of the ordered sample. But the above argument can be saved in the special case when the cube {1, …, n}^{k} can be split into finitely many rectangles where the function a(j_{1}, …, j_{k}) is equal to a constant. Then limit theorems for weighted U-statistics can be proved in cases when the function a(j_{1}, …, j_{k}) can be well approximated by such simple functions. We shall apply this approach, and throughout the proof we heavily exploit the L^{2} isomorphism property of stochastic integrals. We also use Poissonian approximation, a method which helped to overcome certain technical difficulties. The idea that Poissonian approximation is useful for the investigation of U-statistics appeared in the paper of Dynkin and Mandelbaum [1], and we borrowed it from there.

2. Formulation of the main results. In this Section we formulate the main results of this paper. We introduce the following notation: Given a real number x, let [x] denote its integer part. Our first result is the following

Theorem 1. Let U_{n} be defined by formula (1.1) with a function satisfying (1.2) and (1.3). If there is a continuous function A(y_{1}, …, y_{k}) on [0, 1]^{k} such that for A_{n}(y_{1}, …, y_{k}) = a([ny_{1}], …, [ny_{k}]) the relation

lim_{n→∞} ∫_{[0,1]^{k}} |A(y_{1}, …, y_{k}) − A_{n}(y_{1}, …, y_{k})|^{2} dy_{1} … dy_{k} = 0

holds, then the sequence n^{−k/2}U_{n} tends in distribution to the stochastic integral

V = \frac{1}{k!} ∫ f(x_{1}, …, x_{k})A(y_{1}, …, y_{k}) B( dx_{1}, dy_{1}) … B( dx_{k}, dy_{k}) ,

where B(·, ·) is a Wiener sheet.

Let us remark that Theorem 1 is not an empty statement. Its condition can be satisfied for instance if the function a(j_{1}, …, j_{k}) is chosen in such a way that its value depends only on the direction of the vector (j_{1}, …, j_{k}) in ℝ^{k}, and it depends on this direction continuously. The subsequent Theorems 2 and 3 are natural generalizations of the results in Section 4 of [9].

Theorem 2. Let U_{n} be defined by formula (1.1) with a function satisfying (1.2) and (1.3). Assume that a(j_{1}, …, j_{k}) in formula (1.1) can be written in the form

a(j_{1}, …, j_{k}) = u(h(j_{1}), …, h(j_{k})) ,

where h: Z^{1} → {1, …, r} with some integer r is such that the limit

lim_{n→∞} \frac{1}{n}#\{j, j ≤ n, h(j) = s\} = H(s)

exists for all s = 1, …, r, and u is an arbitrary function on {1, …, r}^{k}. Then the sequence n^{−k/2}U_{n} converges in distribution to the stochastic integral

V = \frac{1}{k!} ∫ f(x_{1}, …, x_{k})A(y_{1}, …, y_{k})B( dx_{1}, dy_{1}) … B( dx_{k}, dy_{k}) ,

where B(·, ·) is a Wiener sheet, and

A(y_{1}, …, y_{k}) = u(j_{1}, …, j_{k})
if H(1) + ⋯ + H(j_{s} − 1) < y_{s} ≤ H(j_{1}) + ⋯ + H(j_{s}), 1 ≤ s ≤ k .

Theorem 3. Let us consider a sequence of random variables U_{n} defined by formula (1.1) with a function f satisfying (1.2) and (1.3) and a weight function of the form

a(j_{1}, …, j_{k}) = e(j_{1}) ⋯ e(j_{k})

with a sequence e(j), j = 1, 2, …, such that the sequence e(j) is bounded, and the limit

lim_{n→∞} \frac{1}{n} ∑_{j=1}^{n} e(j)^{2} = E > 0  (2.1)

exists. Then the random variables n^{−k/2}U_{n} converge in distribution to

V = \frac{1}{k!}E^{k/2} ∫ f(x_{1}, …, x_{k})W ( dx_{1}) … W ( dx_{k}) ,

where W (·) is a Wiener process on [0, 1].

Let us remark that, up to a scaling factor, the limit in Theorem 3 is insensitive to the choice of the sequence e(j).

Let us discuss the distribution of the U-statistics (1.1) if relation (1.3) may not hold. We get, by expressing the terms f(X_{j_{1}}, …, X_{j_{k}}) by means of the Hoeffding decomposition (1.4), that

U_{n} = \frac{1}{k!} ∑_{s=0}^{k} ∑_{\begin{matrix}1≤j_{p}≤n, 1≤p≤s \\ and j_{p}≠j_{p′} if p≠p′ \end{matrix}} B_{n}(j_{1}, …, j_{s})f_{s}(X_{j_{1}}, …, X_{j_{s}})  (2.2)

with

B_{n}(j_{1}, …, j_{s}) = ∑_{1≤l_{p}≤n, 1≤p≤k l_{p}≠l_{p′} if p≠p′ j_{p}=l_{rp} for some r_{p} p=1,...,s such that 1≤r_{1}<···<r_{s}≤k} a(l_{1}, …, l_{k}) ,  (2.3)

or by exploiting the symmetry of the function a(l_{1}, …, l_{k})

B_{n}(j_{1}, …, j_{s}) = \binom{k}{s} ∑_{\begin{matrix}1≤l_{p}≤n, 1≤p≤k \\ l_{p}≠l_{p′} if p≠p′ \\ j_{p}=l_{p} for p=1,...,s \end{matrix}} a(l_{1}, …, l_{k}) .  (2.3′)

For unweighted U-statistics B_{n}(j_{1}, …, j_{s}) = \binom{k}{s}(n−s−1) ⋯ (n−k) ≍ \binom{k}{s}n^{k−s}.

The orthogonality of the random variables f_{s}(X_{j_{1}}, …, X_{j_{s}}) together with this relation imply that the the inner sum with the smallest index s for which f_{s} does not vanish identically gives the dominating contribution to the external sum in (2.2), and it has order n^{k−s/2}. For typical weighted U-statistics a similar picture arises. But since the coefficients a(j_{1}, …, j_{k}) may cause some additional cancellation, the situation is more complex. We show this in an example which may be of special interest. We consider the model in Theorem 3, but do not assume that the kernel function f defines degenerate statistics. We consider statistics of the form

U_{n} = ∑_{1≤j_{1}<j_{2}···<j_{k}≤n} e(j_{1}) ⋯ e(j_{k})f(X_{j_{1}}, …, X_{j_{k}}) .  (2.4)

Put

F_{n} = n^{−1/2} ∑_{j=1}^{n} e(j) .  (2.5)

The limit behavior of U_{n} is different in the cases when F_{n} has a finite limit and when it tends to zero or to infinity. We describe the case when F_{n} has a finite limit. This seems to be the most interesting case, when the contributions of different terms in the Hoeffding representation have the same order and the limit can be represented as a sum of stochastic integrals of different multiplicity. This question was considered in a special case in papers [4] and [9], and it also shows some analogy with the surface charge in [6]. The remaining cases will be only briefly discussed.

Theorem 4. Let us consider the weighted U-statistics defined in (2.4) with a bounded sequence e(j), j = 1, 2, …, satisfying (2.1) and a square integrable kernel function f. Assume that the sequence F_{n} defined in (2.5) has a limit lim_{n→∞} F_{n} = F. Let us take the Hoeffding decomposition of the function f given in formulas (1.4) and (1.4′). Then the sequence n^{−k/2}U_{n} converges in distribution to the sum of stochastic integrals

\frac{D_{k}f_{0}}{k!} + ∑_{s=1}^{k} \frac{D_{k−s}}{s!(k − s)!}E^{s/2} ∫ f_{s}(x_{1}, …, x_{s}) W ( dx_{1}) … W ( dx_{s})

as n → ∞, where W (x) is a Wiener process in the interval [0, 1], and the sequence D_{s} is defined by the following recursive formula: D_{0} = 1, D_{1} = F, and

D_{s} = F^{s} − \begin{matrix}[ & \frac{s}{2} & ] \\ ∑ \\ p=1 \end{matrix} \frac{s!}{2^{p}p!(s − 2p)!}E^{p}D_{s−2p} .

3. Approximation of U-statistics. In this section we approximate weighted Ustatistics with polynomials of independent centered Poissonian random variables (by a centered Poissonian random variable we mean a Poissonian random variable minus its expectation) and show that a small error is committed if these centered Poissonian random variables are replaced by independent Gaussian random variables. To formulate these results we introduce some definitions and remarks.

Remark 1. For a function f satisfying (1.1) and any ε > 0 an approximating step-function g(x_{1}, …, x_{k}) = g_{ε}(x_{1}, …, x_{k}) can be given such that

∫_{[0,1]^{k}} |f(x_{1}, …, x_{k}) − g(x_{1}, …, x_{k})|^{2} dx_{1} … dx_{k} < ε ,  (3.1)

and there is some integer L = L(ε) such that the function g(x_{1}, …, x_{k}) is constant

on all cubes (\frac{j_{1} − 1}{L}, \frac{j_{1}}{L} ] × ⋯ × (\frac{j_{k} − 1}{L}, \frac{j_{k}}{L} ], 1 ≤ j_{s} ≤ L for s = 1, …, k, and it

is zero on those cubes for which j_{s} = j_{s′} with some s ≠ s′.

We introduce the notion of ε-approximability of a weight function a(j_{1}, …, j_{k}).

Definition of ε-approximation of weight functions. A sequence a(j_{1}, …, j_{k}) is ε-approximable by a set of elementary functions b_{n}^{ε}(j_{1}, …, j_{k}), 1 ≤ j_{s} ≤ n, 1 ≤ s ≤ k if

n^{−k} ∑ |a(j_{1}, …, j_{k}) − b_{n}^{ε}(j_{1}, …, j_{k})|^{2} < const. ε

with some constant independent of n and ε, and the function b_{n}^{ε}(j_{1}, …, j_{k}) has the following property:

There exists a partition Λ_{1} = Λ_{1}(n, ε), …, Λ_{p} = Λ_{p}(n, ε) of the set {1, …, n} with cardinality |Λ_{1}| = N_{1} = N_{1}(n, ε), …, |Λ_{p}| = N_{p} = N_{p}(n, ε) with some number p = p(ε) which may depend on ε but not on n and numbers B_{n}^{ε}(m_{1}, …, m_{k}) whose absolute values are bounded by some number B(ε) which does not depend on n, 1 ≤ m_{s} ≤ p, s = 1, …, k, such that

b_{n}^{ε}(j_{1}, …, j_{k}) = B_{n}^{ε}(m_{1}, …, m_{k}) if j_{s} ∈ Λ_{m_{s}} for all 1 ≤ s ≤ k .  (3.2)

We shall say that the above ε-approximation is determined at level n by the partition Λ_{1}, …, Λ_{p} of the set {1, …, n} and the function B_{n}^{ε}(m_{1}, …, m_{k}).

Now we formulate the results of this Section.

Lemma 1. Let U_{n} be a weighted U-statistic as defined in (1.1) with a kernel function f satisfying (1.2) and (1.3) and a weight function a(j_{1}, …, j_{k}) which is ε-approximable. Let this ε-approximation be determined at level n by a partition Λ_{1}, …, Λ_{p} of the set {1, …, n} and a function B_{n}^{ε}(m_{1} …, m_{k}). Take an ε-approximating step function g(x_{1}, …, x_{k}) = g_{ε}(x_{1}, …, x_{k}) of the function f which satisfies the properties formulated in Remark 1 and put

g^{∗}(l_{1}, …, l_{k}) = g (\frac{l_{1}}{L} , …, \frac{l_{k}}{L} ) ,  (3.3)

where L is the same as in Remark 1. A set of independent centered Poissonian random variables η_{m,l}, 1 ≤ m ≤ p and 1 ≤ l ≤ L, can be constructed with

parameter \frac{N_{m}}{L} (N_{m} is the cardinality of the set Λ_{m}) such that

E|n^{−k/2}(U_{n} − \frac{1}{k!} ∑_{\begin{matrix}m_{s}=1,...,p \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}} B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, …, l_{k})η_{m_{1},l_{1}} … η_{m_{k},l_{k}})|^{2}

< const. (ε + \frac{C(ε, k)}{\sqrt{n}})

with some constant C(ε, k) depending only on ε and k.

Lemma 2. Let us fix some positive integers p and k. Let us have for all positive integers n a sequence of independent centered Poissonian random variables η_{s} = η_{s}(n), with parameter N_{s} and a sequence of independent Gaussian random variables ξ_{s} = ξ_{s}(n) with expectation zero and variance N_{s}, 1 ≤ s ≤ p, such that N_{s} ≤ n, 1 ≤ s ≤ p. Consider the polynomials

S_{n} = n^{−k/2} ∑_{\begin{matrix}j_{s}=1,...,p \\ j_{s}≠j_{s′} if s≠s′ \end{matrix}} b_{n}(j_{1}, …, j_{k}) η_{j_{1}} ⋯ η_{j_{k}}

and

T_{n} = n^{−k/2} ∑_{\begin{matrix}j_{s}=1,...,p \\ j_{s}≠j_{s′} if s≠s′ \end{matrix}} b_{n}(j_{1}, …, j_{k}) ξ_{j_{1}} ⋯ ξ_{j_{k}}

with coefficients satisfying the relation

|b_{n}(j_{1}, …, j_{k})| < K for all 1 ≤ j_{s} ≤ p, 1 ≤ s ≤ k

with some positive constant K. Then for all t ∈ ℝ^{1}

lim_{n→∞} (Ee^{itS_{n}} − Ee^{itT_{n}}) = 0 .

Proof of Lemma 1. Introduce the expression

U_{n}^{(1)} = ∑_{1≤j_{1}<j_{2}···<j_{k}≤n} b_{n}^{ε}(j_{1}, …, j_{k})f(X_{j_{1}}, …, X_{j_{k}}) ,

where b_{n}^{ε} is the ε-approximating sequence of a_{n}. Since

Ef(X_{j_{1}}, …, X_{j_{k}})f(X_{j′_{1}}, …, X_{j′_{k}}) = 0 if (j_{1}, …, j_{k}) ≠ (j′_{1}, …, j′_{k})

by (1.3), hence

En^{−k}(U_{n} − U_{n}^{(1)})^{2} = n^{−k} ∑ |a(j_{1}, …, j_{k}) − b_{n}^{ε}(j_{1}, …, j_{k})|^{2} Ef^{2}(X_{j_{1}}, …, X_{j_{k}})
< const. ε.  (3.4)

Let ν_{1}, …, ν_{p} be independent Poisson distributed random variables with parameter N_{s}, 1 ≤ s ≤ p, independent of the random variables X_{j}, j = 1, …, n, too. Let the sets Λ_{m} appearing in the definition of ε-approximability be Λ_{m} = {s_{1}(m), …, s_{N_{m}}(m)}, s_{1}(m) < ⋯ < s_{N_{m}}(m), m = 1, …, p. We define sets Λ′_{m}, with (random) size ν_{m}, 1 ≤ m ≤ p, which are close to the sets Λ_{m}. Put Λ′_{m} = {s̄_{1}(m), …, s̄_{ν_{m}}(m)}, s̄_{1}(m) < ⋯ < s̄_{ν_{m}}(m) such that s̄_{l}(m) = s_{l}(m)

for l ≤ min(N_{m}, ν_{m}) and s̄_{l}(m) = J(m) + l − N_{m} with J(m) = ∑_{p=1}^{m−1}(ν_{p} − N_{p})_{+} if

N_{m} < l ≤ ν_{m}. We consider a set of independent random variables Y_{m,l}, 1 ≤ m ≤ p and 1 ≤ l ≤ ν_{m}, with uniform distribution in the interval [0, 1] which are independent of the random variables ν_{m}, n = 1, …, p, and also have the property:

Y_{m,l} = X_{s_{l}(m)} if l ≤ min(ν_{m}, N_{m}), m = 1, …, p .

(The choice of the random variables Y_{m,l} with such properties is possible. They must be chosen conditionally independent and uniformly distributed on [0, 1] under the condition that the values of the random variables ν_{m} are prescribed.) Define the numbers l(j) and m(j) as the indices such that j ∈ Λ′_{m(j)} and j = s̄_{l(j)}(m(j)) if j is an element of some Λ′_{m}, 1 ≤ m ≤ p. Otherwise let l(j) and m(j) be meaningless. For 1 ≤ m_{i} ≤ p and l_{i} = 1, 2, …, i = 1, …, k put

b̄_{n}^{ε} ((m_{1}, l_{1}), …, (m_{k}, l_{k})) = \{ \begin{matrix}B_{n}^{ε}(m_{1}, …, m_{k}) if l_{i} ≤ ν_{m_{i}}, i = 1, …, k \\ 0 otherwise \end{matrix}

and

U_{n}^{(2)} = ∑_{1≤j_{1}<j_{2}···<j_{k}<∞}b̄_{n}^{(ε)} ((m(j_{1}), l(j_{1})), …, ((m(j_{k}), l(j_{k})))
× f (Y_{m(j_{1}),l(j_{1})}, …, Y_{m(j_{k}),l(j_{k})}) .

We define b̄_{n}^{(ε)}((m(j_{1}), l(j_{1})) …, ((m(j_{k}), l(j_{k}))) = 0 in the last expression if m(j_{s}) and l(j_{s}) are not defined for some s. We claim that

E (n^{−k/2}(U_{n}^{(1)} − U_{n}^{(2)}))^{2} < C(ε, k)n^{−1/2} .  (3.5)

To prove relation (3.5) observe that the number of terms which appear in the sum U_{n}^{(1)} but not in U_{n}^{(2)} (two terms in these sums agree if the function f is a function of the same random variables in them, and it has the same coefficient) and the number of terms which appear in U_{n}^{(2)}, but not in U_{n}^{(1)} is less than

k max_{s≤p} |ν_{s} − N_{s}| (max_{s≤p} N_{s}^{k−1} + max_{s≤p} ν_{s}^{k−1}) .

This relation together with the orthogonality relations imply that

E (n^{−k/2}(U_{n}^{(1)} − U_{n}^{(2)}))^{2} < \frac{C(ε, k)}{n^{k}}E (max_{s≤p} |ν_{s} − N_{s}| (max_{s≤p} N_{s}^{k−1} + max_{s≤p} ν_{s}^{k−1})) .

(3.6)

Since

Eν_{s}^{L} ≤ C(L)N_{s}^{L} ≤ C(L)n^{L}

and

E|ν_{s} − N_{s}|^{2} ≤ n

with some C(L) > 0 for any s ≤ p and L ≥ 1, hence by the Schwartz inequality

[E (max_{s≤p} |ν_{s} − N_{s}| (max_{s≤p} N_{s}^{k−1} + max_{s≤p} ν_{s}^{k−1})]^{2}
≤ E max |ν_{s} − N_{s}|^{2} · E [max N_{s}^{k−1} + max ν_{s}^{k−1}]^{2} ≤ const. n^{2k−1}.

The last inequality together with (3.6) imply (3.5).

Put Σ = Σ(ε) = [0, 1] × {1, …, p}, and define the random field consisting of the points Z(m, l) = (Y_{m,l}, m), 1 ≤ m ≤ p and 1 ≤ l ≤ ν_{m} on it. (Σ depends on ε through p = p(ε).) Then Z(m, l) is a Poisson process such that the expected

value of the points Z(·, ·) in a set ⋃_{m=1}^{p}(A_{m}, m) ⊂ Σ equals ∑_{m=1}^{p} N_{m}λ(A_{m}), where

λ(·) denotes the Lebesgue measure. Introduce the counting measure μ_{n} = μ_{n}^{ε} on Σ such that μ_{n}(B) is the number of points Z(·, ·) in the set B for B ⊂ Σ. Let P_{n} be its centering, i.e. P_{n}(B) = μ_{n}(B) − Eμ_{n}(B). Given a function f(x_{1}, …, x_{k}) on [0, 1]^{k} define the function f ε ; b̄((x_{1}, m_{1}), …, (x_{k}, m_{k})) on Σ^{k} as

f ε ; b̄((x_{1}, m_{1}), …, (x_{k}, m_{k})) = B_{n}^{ε}(m_{1}, …, m_{k})f(x_{1}, …, x_{k}) ,

where the function B_{n}^{ε} is the same as that which appears in the definition of ε-approximability of a weight function. Then U_{n}^{(2)} can be rewritten as

U_{n}^{(2)} = \frac{1}{k!} ∫′_{Σ^{k}} f ε ; b̄(z_{1}, …, z_{k}) μ_{n}(dz_{1}) … μ_{n}(dz_{k})

with z_{s} = (x_{s}, m_{s}), x_{s} ∈ [0, 1] and m_{s} ∈ {1, …, p} for s = 1, …, k, where ∫′ means that the hyperplanes z_{j} = z_{j′} for j ≠ j′ are cut out from the domain of integration. Condition (1.3) also implies that

∫_{Σ} f ε ; b̄(z, z_{2}, …, z_{k}) λ̄(dz) = 0 for all z_{2}, …, z_{k}

with λ̄(A) = Eμ_{n}(A) for A ⊂ Σ. Hence

U_{n}^{(2)} = \frac{1}{k!} ∫′_{Σ^{k}} f ε ; b̄(z_{1}, …, z_{k}) P_{n}(dz_{1}) … P_{n}(dz_{k}) .  (3.7)

Define the mapping I from the set of function f ε ; b̄ to the space of random variables on (Ω, A, P), where (Ω, A, P) is the probability space where the Poisson process is defined, as

I(f ε ; b̄) = \frac{1}{\sqrt{k!}} ∫′_{Σ^{k}} f ε ; b̄(z_{1}, …, z_{k}) P_{n}(dz_{1}) … P_{n}(dz_{k}) .

It is known in the theory of Poissonian integrals, and actually it is not difficult to prove that

∫ f ε ; b̄(z_{1}, …, z_{k})^{2} λ̄(dz_{1}) … λ̄(dz_{k}) = EI(f ε ; b̄)^{2} .

Let g(x_{1}, …, x_{k}) = g_{ε}(x_{1}, …, x_{k}) be an approximating function of f having the properties mentioned in Remark 1. Since N_{s} ≤ n for all 1 ≤ s ≤ p, λ̄(A) ≤ nλ(A) for A ⊂ Σ, where λ(·) denotes the Lebesgue measure on Σ. This fact together with (3.1) and the definition of gε ; b̄ imply that

∫ |f ε ; b̄(z_{1}, …, z_{k}) − gε ; b̄(z_{1}, …, z_{k})|^{2} λ̄( dz_{1}) … λ̄( dz_{k}) < const. εn^{k} .

The last relation together with (3.7) and the L^{2} isomorpism of the mapping I (applying it for f − g) imply that

n^{−k}E [U_{n}^{(2)} − \frac{1}{k!} ∫′_{Σ^{k}} gε ; b̄(z_{1}, …, z_{k}) P_{n}(dz_{1}) … P_{n}(dz_{k})]^{2} ≤ const. ε .

This relation together with (3.4) and (3.5) give that

n^{−k}E [U_{n} − \frac{1}{k!} ∫′_{Σ^{k}} gε ; b̄(z_{1}, …, z_{k}) P_{n}(dz_{1}) … P_{n}(dz_{k})]^{2} ≤ (const. ε + \frac{C(ε, k)}{\sqrt{n}}) .

(3.8)

The random measure P_{n} ((\frac{l − 1}{L}, \frac{l}{L}] , m) is a centered Poissonian random

variable with parameter \frac{N_{m}}{L}, and the measures of the sets ((\frac{l − 1}{L}, \frac{l}{L}] , m) are

independent for different pairs (l, m). Hence

∫′_{Σ^{k}} gε ; b̄(z_{1}, …, z_{k}) P_{n}(dz_{1}) … P_{n}(dz_{k})

= ∑_{\begin{matrix}m_{s}=1,...,p \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}} B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, …, l_{k})η_{m_{1},l_{1}} ⋯ η_{m_{k},l_{k}} ,

and relation (3.8) implies Lemma 1. □

Proof of Lemma 2. Since

|exp (i ∑ a_{j}) − exp (i ∑ b_{j})| ≤ ∑ | exp(ia_{j}) − exp(ib_{j})| ,

hence

|E exp{itS_{n}} − E exp{itT_{n}}|
≤ const. sup_{\begin{matrix}|s|≤K|t| \\ j_{1}...,j_{k} \end{matrix}} |E exp \{is\frac{η_{j_{1}}}{\sqrt{n}} ⋯ \frac{η_{j_{k}}}{\sqrt{n}}\} − E exp \{is\frac{ξ_{j_{1}}}{\sqrt{n}} ⋯ \frac{ξ_{j_{k}}}{\sqrt{n}}\}| (3.9)

with some K > 0. We may assume that

sup_{j≤p} E |n^{−1/2}(η_{j}(n) − ξ_{j}(n))|^{2} → 0 as n → ∞ .  (3.10)

Indeed, if \frac{η_{j}}{\sqrt{N_{j}}} is the quantile transform of \frac{ξ_{j}}{\sqrt{N_{j}}}, i.e.

\frac{η_{j}}{\sqrt{N_{j}}} = F_{j}^{−1} (Φ (\frac{ξ_{j}}{\sqrt{N_{j}}} )) ,

where Φ is the standard normal distribution function, F_{j} is the distribution function of \frac{η_{j}}{\sqrt{N_{j}}} and N_{j} is the variance of ξ_{j} and η_{j}, then it is not difficult to see with

the help of the central limit theorem that (3.10) holds for this ξ_{j} and η_{j}. (Actually the following stonger estimate holds. See formula (2.6) in Lemma 1 of [5].)

E |n^{−1/2}(η − ξ)|^{2} ≤ const. \frac{1}{n} .

On the other hand, the random variable S_{n} defined with these random variables η_{j} has the right distribution. Then we have

|E exp \{is\frac{η_{j_{1}}}{\sqrt{n}} ⋯ \frac{η_{j_{k}}}{\sqrt{n}}\} − E exp \{is\frac{ξ_{j_{1}}}{\sqrt{n}} ⋯ \frac{ξ_{j_{k}}}{\sqrt{n}}\}|

≤ n^{−k/2}|s| E |η_{j_{1}} ⋯ η_{j_{k}} − ξ_{j_{1}} ⋯ ξ_{j_{k}}|

≤ n^{−k/2}|s| ∑_{p=0}^{k−1} E |η_{j_{1}} ⋯ η_{j_{p}} | |η_{j_{p+1}} − ξ_{j_{p+1}} | |ξ_{j_{p+2}} ⋯ ξ_{j_{k}} |

= n^{−k/2}|s| ∑_{p=0}^{k−1} E |η_{j_{1}}| ⋯ E |η_{j_{p}} | E |η_{j_{p+1}} − ξ_{j_{p+1}} | E |ξ_{j_{p+2}} | ⋯ E |ξ_{j_{k}}|

≤ n^{−1/2}|s| const. ∑_{p=0}^{k−1} E |η_{j_{p+1}} − ξ_{j_{p+1}} |

≤ const. sup E |n^{−1/2}(η_{j}(n) − ξ_{j}(n))|^{2}

because of the independence of the pairs (η_{j}(n), ξ_{j}(n)) and the condition N_{j} ≤ n. The last relation together with (3.10) imply that the right-hand side of (3.9) tends to zero, hence Lemma 2 holds. □

4. Proof of the Theorems.

Proof of Theorem 1. There is a step function A^{ε}(y_{1}, …, y_{k}) such that

∫_{[0,1]^{k}} |A_{n}(y_{1}, …, y_{k}) − A^{ε}(y_{1}, …, y_{k})|^{2} dy_{1} …, dy_{k} < ε

for n > n(ε), and it has the following structure: There is some T > 0 such that

A(y_{1}, …, y_{k}) = A^{ε} (\frac{m_{1}}{T} , …, \frac{m_{k}}{T} ) if \frac{m_{s} − 1}{T} < y_{s} ≤ \frac{m_{s}}{T}

and all numbers m_{1}, …, m_{k} are different

= 0 if there is some 1 ≤ s < s′ ≤ k and 0 < m ≤ T

such that \frac{m − 1}{T} < y_{s}, y′_{s} ≤ \frac{m}{T} .

There is an ε-approximation of the function a(m_{1}, …, m_{k}) which is determined at

level n > n(ε) by the partition Λ_{m} = ([\frac{m − 1}{T}n] , [\frac{m}{T}n]], 1 ≤ m ≤ T, and the

functions

B_{n}^{ε}(m_{1}, …, m_{k}) = A^{ε} (\frac{m_{1}}{T} , …, \frac{m_{k}}{T} ) .

Let g(x_{1}, …, x_{k}) = g_{ε}(x_{1}, …, x_{k}) be an ε-approximating step function of f which satisfies Remark 1. Let the function g^{∗}(l_{1}, …, l_{k}) be defined by (3.3) and the above function g. We get by Lemma 1 that for

S_{n} = \frac{1}{k!}n^{−k/2} ∑_{\begin{matrix}m_{s}=1,...,T \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}} B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, …, l_{k})η_{m_{1},l_{1}} … η_{m_{k},l_{k}}

E(n^{−k/2}U_{n} − S_{n})^{2} ≤ const. (ε + \frac{C(ε, k)}{\sqrt{n}}) ,  (4.1)

where η_{m,l}, 1 ≤ m ≤ T and 1 ≤ l ≤ L, are appropriate independent centered Poissonian random variables with parameter \frac{n}{T}.

On the other hand,

∫[0,1]^{2k} |f(x_{1}, …, x_{k})A(y_{1}, …, y_{k}) − g_{ε}(x_{1}, …, x_{k})A^{ε}(y_{1}, …, y_{k})|^{2}
dx_{1} dy_{1} … dx_{k} dy_{k} ≤ const. ε ,

and because of the L^{2} isomorphism of Wiener-Itô integrals

E (V − T_{n})^{2} ≤ const. ε ,  (4.2)

where V is the stochastic integral with the limit distribution defined in the formulation of Theorem 1, and

T_{n} = \frac{1}{k!}n^{−k/2} ∑_{\begin{matrix}m_{s}=1,...,T \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}} B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, …, l_{k})ξ_{m_{1},l_{1}} … ξ_{m_{k},l_{k}}

with independent Gaussian random variable ξ_{m,l}, 1 ≤ m ≤ T and 1 ≤ l ≤ L, with expectation zero and variance \frac{n}{T}.

It follows from (4.1) that

|E exp{itn^{−k/2}U_{n}} − E exp{itS_{n}}| ≤ |t|E|n^{−k/2}U_{n} − S_{n}|
≤ |t| (E(n^{−k/2}U_{n} − S_{n})^{2})^{1/2} ≤ const. (ε^{1/2} + C(ε, k)n^{−1/4})

for any t ∈ ℝ^{1}. Similarly, it follows from (4.2) that

|Ee^{itV} − Ee^{itT_{n}} | ≤ const. ε^{1/2} .

Since Ee^{itS_{n}} − Ee^{itT_{n}} → 0 by Lemma 2 the last two relations imply that

lim sup_{n→∞} |E exp{itn^{−k/2}U_{n}} − E exp{itV }| ≤ const. ε^{1/2}

Since the last relation holds for any ε > 0 we get that the characteristic function of U_{n} satisfies the relation

E exp{itn^{−k/2}U_{n}} → E exp{itV } for all t ∈ ℝ^{1} .

The last relation implies Theorem 1. □

Proof of Theorem 2. The proof is similar to that of Theorem 1. Now we can choose the function a(j_{1}, …, j_{k}) itself as its approximation by elementary function. Then this approximation is determined at level n by the sets

Λ_{m} = {j; 1 ≤ j ≤ n, h(j) = m} m = 1, …, r ,

and the function B_{n}^{ε}(m_{1}, …, m_{k}) = u(m_{1}, …, m_{k}). Then N_{m} = N_{m}(n), the cardinality of the set Λ_{m}, satisfies the relation

lim_{n→∞} \frac{N_{m}(n)}{n} = H(m) for m = 1, …, r .  (4.3)

Let g = g_{ε} be an approximating step function of f satisfying Remark 1, and let the function g^{∗} be defined by (3.3). Then

S_{n} = \frac{1}{k!}n^{−k/2} ∑_{\begin{matrix}m_{s}=1,...,r \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}} B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, ⋯, l_{k})η_{m_{1},l_{1}} … η_{m_{k},l_{k}}

well approximates n^{−k/2}U_{n} in L^{2} norm, where η_{m,l} are independent centered Pois-

sonian random variables with parameter \frac{N_{m}}{L}. Because of the definition of the function A(y_{1}, …, y_{k}) and (4.3) the stochastic integral V appearing in Lemma 2 can be well approximated in L^{2} norm by

T_{n} = \frac{1}{k!}n^{−k/2} ∑_{\begin{matrix}m_{s}=1,...,r \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}} B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, …, l_{k})ξ_{m_{1},l_{1}} ⋯ ξ_{m_{k},l_{k}} ,

where ξ_{m,l} are independent Gaussian random variables with expectation zero and

variance \frac{N_{m}}{L}. Then Lemma 2 implies that the characteristic functions of S_{n} and T_{n} are close to each other. Then a natural adaptation of the argument in the proof of Theorem 1 implies that the characteristic function of n^{−k/2}U_{n} tends to that of V, and this implies Theorem 2. □

In the proof of Theorem 3 we need a lemma which shows why the sequence e(j) influences only the norming constant of the limit distribution of U_{n} in Theorem 3.

Lemma 3. Let f(x_{1}, …, x_{k}) be a square integrable function on [0, 1]^{k}, h(y) a function on [0, 1] such that ∫_{0}^{1} h^{2}(y) dy = 1, W (x) a Wiener process on [0, 1] and B(x, y) a Wiener sheet on [0, 1]^{2}. Then the stochastic integrals

I_{1} = ∫ f(x_{1}, …, x_{k}) W ( dx_{1}) … W ( dx_{k})

and

I_{2} = ∫ f(x_{1}, …, x_{k})h(y_{1}) ⋯ h(y_{k}) B( dx_{1}, dy_{1}) … B( dx_{k}, dy_{k})

have the same distribution.

Proof of Lemma 3. This lemma could have been proved by considering first elementary functions and then approximating general functions by them. We choose a different way. We express both I_{1} and I_{2} by means of Itô’s formula as a series of independent Gaussian random variables and observe that these two expressions have the same distribution.

Let ψ_{1}, ψ_{2}, … be a complete orthonormal system in [0, 1], and take the expansion

f(x_{1}, …, x_{k}) = ∑ c(j_{1}, …, j_{k})ψ_{j_{1}}(x_{1}) ⋯ ψ_{j_{k}}(x_{k}) .

The functions φ_{j}(x, y) = ψ_{j}(x)h(y), j = 1, 2, …, are orthonormal in [0, 1]^{2}, and

f(x_{1}, …, x_{k})h(y_{1}) ⋯ h(y_{k}) = ∑ c(j_{1}, …, j_{k})φ_{j_{1}}(x_{1}, y_{1}) ⋯ φ_{j_{k}}(x_{k,}, y_{k}) .

By Itô’s formula (see [3], or [7], Section 7) these relations imply that

I_{1} = ∑ c(j_{1}, …, j_{k}) :η_{j_{1}} ⋯ η_{j_{k}}:  (4.4)

and

I_{2} = ∑ c(j_{1}, …, j_{k}) :ζ_{j_{1}} ⋯ ζ_{j_{k}}:  (4.4′)

with η_{j} = ∫ ψ(x) W ( dx) and ζ_{j} = ∫ φ(x, y) B (dx, dy). Here :η_{j_{1}} ⋯ η_{j_{k}}:, the Wick polynomial of the corresponding product, equals ∏ H_{l_{m}}(η_{m}), where l_{m} denotes the multiplicity of the index m in the set {j_{1}, …, j_{k}} and H_{m}(x) is the m-th Hermite polynomial. The definition of :ζ_{j_{1}} ⋯ ζ_{j_{k}}: is similar. Since both sequences η_{j} and ζ_{j}, j = 1, 2, …, are sequences of independent standard normal random variables, the expressions in (4.4) and (4.4′) have the same distributions. Lemma 3 is proved. □

Proof of Theorem 3. The proof is similar to that of Theorems 1 and 2. Let us fix some small ε > 0, and define the sequence ē(j) = ē^{ε}(j), j = 1, 2, …, by the formula

ē(j) = Kε, if Kε ≤ e(j) < (K + 1)ε with some integer K .

Then

|\frac{1}{n} ∑_{j=1}^{n} ē^{2}(j) − \frac{1}{n} ∑_{j=1}^{n} e^{2}(j)| ≤ const. ε ,  (4.5)

and the sequence ē(j), j = 1, 2, … takes finitely many values K_{1}ε < K_{2}ε < ⋯ < K_{p}ε with some p = p(ε) because of the boundedness of the sequence e(j). Let the sequence ē(j), j = 1, …, n, take the value K_{l}ε N_{l} = N_{l}(n) times, 1 ≤ l ≤ p. Introduce the function h_{n}(y) = h_{n}^{ε}(y) on [0, 1] as

h_{n}(y) = K_{p}ε on the interval \frac{1}{n} ∑_{l=1}^{p−1} N_{l} < y ≤ \frac{1}{n} ∑_{l=1}^{p} N_{l}

and the number E(n) = E^{ε}(n) = ∫_{0}^{1} h_{n}^{2}(y) dy. By Lemma 3 the stochastic integral

V_{n} = \frac{1}{k!} (\frac{E}{E(n)})^{k/2}∫ f(x_{1}, …, x_{k})h_{n}(y_{1}) ⋯ h_{n}(y_{k})B( dx_{1}, dy_{1}) … B( dx_{k}, dy_{k})

(4.6)

has the same distribution as the stochastic integral V defined in the formulation of Theorem 3.

The sequence a(j_{1}, …, j_{k}) = e(j_{1}) ⋯ e(j_{k}) can be ε-approximated by elementary functions such that this approximation is determined at level n by the partition

Λ_{m} = {j; 1 ≤ j ≤ n, ē(j) = K_{m}ε} for 1 ≤ m ≤ p

and the function B_{n}^{ε}(m_{1}, …, m_{k}) = K_{m_{1}}ε ⋯ K_{m_{k}}ε.

Let g_{ε}(x_{1}, …, x_{k}) = g(x_{1}, …, x_{k}) be an approximating step function of f satisfying Remark 1. Then the random variables n^{−k/2}U_{n} can be well approximated in L^{2} norm by

S_{n} = \frac{1}{k!}n^{−k/2} ∑_{\begin{matrix}m_{s}=1,...,p \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}} B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, …, l_{k})η_{m_{1},l_{1}} … η_{m_{k},l_{k}}

by Lemma 1, where η_{m,l} are independent centered Poissonian random variables

with parameter \frac{N_{m}}{L}. Because of the L^{2} isomorphism property of Wiener-Itô in-

tegrals the random variable V_{n} defined in (4.6) is well approximated in L^{2} norm by

T_{n} = \frac{1}{k!} (\frac{E}{E(n)})^{k/2}n^{−k/2} ∑_{\begin{matrix}m_{s}=1,...,p \\ l_{s}=1,...,L \\ for s=1,...,k \end{matrix}}B_{n}^{ε}(m_{1}, …, m_{k})g^{∗}(l_{1}, …, l_{k})ξ_{m_{1},l_{1}} … ξ_{m_{k},l_{k}} ,

where ξ_{m,l} are independent Gaussian random variables with expectation zero and

variance \frac{N_{m}}{L}. Since

lim_{ε→0} sup_{n} |E^{ε}(n) − E| = 0

by (4.5), Lemma 2 implies that the characteristic functions of S_{n} and T_{n} are close to each other. These relations together with the observation that V_{n} and V have the same distribution, and this implies the proof of Theorem 3 similarly to the proof of Theorem 1. □

The proof of Theorem 4 is based on the following multidimensional version of Theorem 3 and a lemma about the asymptotic behavior of the expression B_{n}(j_{1}, …, j_{k}) defined in (2.3).

Theorem 3′. Consider the random variables

U_{n}^{(s)} = \frac{1}{k!} ∑_{\begin{matrix}1≤j_{p}≤n, 1≤p≤s \\ and j_{p}≠j_{p′} if p≠p′ \end{matrix}} e(j_{1}) ⋯ e(j_{s})f_{s}(X_{j_{1}}, …, X_{j_{s}}), 1 ≤ s ≤ k ,

with a sequence e(j) satisfying (2.1), degenerate functions f_{s}(x_{1}, …, x_{s}), s = 1, …, k, and iid. random variables X_{1}, X_{2}, … with uniform distribution in [0, 1]. The joint distribution of the random variables n^{−s/2}U_{n}^{(s)}, 1 ≤ s ≤ k, tends to that of the random vector

V^{(s)} = \frac{1}{k!}E^{s/2} ∫ f_{s}(x_{1}, …, x_{s}) W ( dx_{1}) … W ( dx_{s}) , 1 ≤ s ≤ k ,

as n → ∞, where W (x) is a Wiener process on [0, 1].

Proof of Theorem 3‘. The proof goes on the same line as that of Theorem 3, only we need a multidimensional version of Lemmas 1, 2 and 3. We only explain the modified Lemmas we need during the proof. The proof of Lemma 3 also yields that if the functions f_{s}(x_{1}, …, x_{s}), 1 ≤ s ≤ k are square integrable and ∫ h^{2}(y) dy = 1, then the joint distribution of the vectors

I_{1}^{(s)} = ∫ f_{s}(x_{1}, …, x_{s}) W ( dx_{1}) … W ( dx_{s}) , 1 ≤ s ≤ k ,

and

I_{2}^{(s)} = ∫ f_{s}(x_{1}, …, x_{s})h(y_{1}) ⋯ h(y_{s}) B( dx_{1}, dy_{1}) … B( dx_{s}, dy_{s}) , 1 ≤ s ≤ k ,

agree.

We need a multidimensional version of Lemma 1, where we have to approximate the sums

U_{n}^{(s)} = ∑_{1≤<j_{1}<j_{2}···<j_{s}≤n} a_{s}(j_{1}, …, j_{s})f_{s}(X_{j_{1}}, …, X_{j_{s}}) 1 ≤ s ≤ k ,

simultaneously if the functions f_{s} satisfy (1.4′), and the sequences a_{s}(j_{1}, …, j_{s}) are all ε-approximable by a set of elementary functions. We want to get the same approximation of the random variables U_{n}^{(s)} as in Lemma 1 for all 1 ≤ s ≤ k (by replacing k by s everywhere) with the following additional restriction: The approximating sums must be the polynomials of the same independent centered Poissonian random variables η_{j,l} for all 1 ≤ s ≤ k. This is possible if the following conditions are satisfied. The ε-approximation of the function a_{s} is determined at level n by a partition Λ_{1}, …, Λ_{p} of {1, …, n} independent of s together with some function B_{n,s}^{ε}(m_{1}, …, m_{p}), and the functions f_{s} are ε-approximated by such step functions g_{s}^{ε}(x_{1}, …, x_{k}) which satisfy Remark 1 with the same constant L in it for all 1 ≤ s ≤ k. These conditions can be satisfied. If the ε-approximation of the function a_{s} is determined at level n by a partition L_{s} = {Λ_{1}(s), …, Λ_{p(s)}(s)} of {1, …, n} which depends on s and some function B_{n,s}^{ε}, 1 ≤ s ≤ k, then it is also determined by a partition which is a refinement of all partitions L_{s}, 1 ≤ s ≤ k, and a function B_{n,s}^{ε} such that relation (3.2) remains valid on the new partition with the same function b_{n}^{ε}(j_{1}, …, j_{s}) = b_{n,s}^{ε}(j_{1}, …, j_{s}). To see that the conditions of Remark 1 can be satisfied simultaneously for all f_{s}, 1 ≤ s ≤ k, observe first that the functions f_{s} can be well approximated in L^{2} norm by continuous functions. This implies that Remark 1 can be satisfied for all sufficiently large L. Then the proof of Lemma 1 can be carried out to supply the strengthened form of Lemma 1 needed for us.

Finally we need the following modified version of Lemma 2. In Lemma 2 we took a polynomial of order k of independent Gaussian and centered Poissonian random variables, and showed that their characteristic functions are close to each other under certain conditions. Take the polynomials of order s for all 1 ≤ s ≤ k of the same random variables, and assume that these polynomials satisfy the conditions of Lemma 2. Consider the random vectors which we get when the centered Poissonian and when the Gaussian random variables are chosen as the arguments of these polymomials. Then the characteristic functions of these random vectors are close to each other. This statement can be proved in the same way as Lemma 2, and Theorem 3′ can be proved by means of these generalized lemmas just as Theorem 3. □

Lemma 4. Let the function B_{n}(j_{1}, …, j_{s}), 1 ≤ s ≤ k, be defined by (2.3) or (2.3′) with a function of the form a(j_{1}, …, j_{k}) = e(j_{1}) ⋯ e(j_{k}). Assume that e(j), j = 1, 2, …, is a bounded sequence satisfying (2.1) and such that lim_{n→∞} F_{n} = F for the sequence F_{n} defined in (2.5). Then

n^{−(k−s)/2}B_{n}(j_{1}, …, j_{s}) = \binom{k}{s}D_{k−s}e(j_{1}) ⋯ e(j_{s}) + ε_{n}^{(s)}(j_{1}, …, j_{s})  (4.7)

such that

lim_{n→∞} sup_{1≤s≤k} sup_{1≤j_{1},...,j_{s}≤n} ε_{n}^{(s)}(j_{1}, …, j_{s}) = 0 ,  (4.7′)

and the sequence D_{s} is defined by the recursive formula D_{0} = 1, D_{1} = F, and

D_{s} = F^{s} − \begin{matrix}[ & \frac{s}{2} & ] \\ ∑ \\ p=1 \end{matrix} \frac{s!}{2^{p}p!(s − 2p)!}E^{p}D_{s−2p} .  (4.8)

Proof of Lemma 4. By formula (2.3′)

B_{n}(j_{1}, …, j_{s}) = \binom{k}{s}G_{n}(j_{1}, …, j_{s})e(j_{1}) ⋯ e(j_{s})  (4.9)

with

G_{n}(j_{1}, …, j_{s}) = ∑_{\begin{matrix}l_{p}∈\{1,...,n\}\\{j_{1},...,j_{s}\}, 1≤p≤k−s \\ l_{p}≠l_{p′} if p≠p′ \end{matrix}} e(l_{1}) ⋯ e(l_{k−s}) .  (4.9′)

We need a good asymptotics for the term G_{n} defined in (4.9′). For this aim we introduce some notations. Given a finite set A let |A| denote its cardinality. For a set U ⊂ {1, …, k} let U_{U} denote the set of all partitions of the set U, and for a set J ⊂ {1, …, n} and a partition (V_{1}, …, V_{p}) of U ⊂ {1, …, k} put

H_{U,J}^{(n)}(V_{1}, …, V_{p}) = ∑_{j_{s}∈{1,...,n}\J, s∈U j_{s}=j_{s′} if j_{s}∈V_{r}, j_{s′} ∈V_{r} for the same 1≤r≤p j_{s}≠j_{s′} if j_{s}∈V_{r}, j_{s′} ∈V_{r′} for r≠r′} ∏_{s∈U} e(j_{s}) .

Let us observe that H_{U,J}^{n}(V_{1}, …, V_{p}) depends only on the cardinalities |V_{1}|, …, |V_{p}| but not on the exact form of the sets V_{1}, …, V_{p}. We claim that if |J| ≤ K with some fixed K > 0, then

|H_{U,J}^{(n)}(V_{1}, …, V_{p})| < const. n^{|U|/2}  (4.10)

and

|H_{U,J}^{(n)}(V_{1}, …, V_{p})| < const. n^{(|U|−1)/2} if |V_{r}| ≥ 3 for some 1 ≤ r ≤ p . (4.11)

We prove (4.10) by induction for the number of elements of the partitions. It holds if the partition consists only of one elements, since

| ∑_{j∈\{1,...,n\}\J} e(j)^{|U|}| < \{ \begin{matrix}const. \sqrt{n} if |U| = 1 \\ const. n if |U| ≥ 2 \end{matrix}  (4.12)

Then relation (4.10) follows from the inductive hypothesis and the identity

\begin{matrix}H_{U,J}^{(n)}(V_{1}, …, V_{p}) = & H_{U\V_{1},J}^{(n)}(V_{2}, …, V_{p}) ∑_{j_{s}∈\{1,...,n\}\J for s∈V_{1}} e(j_{s})^{|V_{1}|} \\ − ∑_{i=2}^{p} H_{U,J}^{(n)}(V_{2}, …, V_{1} ∪ V_{i}, …, V_{p}) . \end{matrix} (4.13)

It is enough to prove (4.11) in the case when |V_{1}| ≥ 3. We can prove it similarly to the relation (4.10) by induction for the number of elements of the partition. If the partition consists of one element, then (4.11) holds because of (4.12), and if it contains more than one element, then it follows from the inductive hypothesis, (4.13), (4.12) and (4.10).

To investigate those partitions of a set U which consist of sets with cardinality one or two we introduce the quantities:

H_{J}^{(n)}(r, s) = H_{U,J}^{(n)}({1, 2}, …, {2r − 1, 2r}, {2r + 1}, …, {2r + s})
with U = {1, …, 2r + s} .

For J = ∅ put

H^{(n)}(r, s) = H_{∅}^{(n)}(r, s) .

We claim that

|H_{J}^{(n)}(r, s) − n^{r}E_{n}^{r}H^{(n)}(0, s)| < const. n^{(2r+s−1)/2}  (4.14)

if |J| ≤ K with some K > 0, where E_{n} = \frac{1}{n} ∑_{j=1}^{n} e(j)^{2}. Relation (4.14) also holds

with s = 0 (and r ≥ 1) if we define H_{n}^{(0)}(0, 0) = 1.

To prove (4.14) observe that

n^{r}E_{n}^{r}H^{(n)}(0, s) = ∑_{j_{u}∈{1,...,n} for 1≤u≤2r+s j_{2u−1}=j_{2u} for 1≤u≤r j_{u}≠j_{u′} if 2r<u, u′≤2r+s and u≠u′} e(j_{1}) ⋯ e(j_{2r+s}) .  (4.15)

Hence

|H_{J}^{(n)}(r, s) − n^{r}E_{n}^{s}H^{(n)}(0, s)| ≤ Σ_{1} + Σ_{2}

with

Σ_{1} = [ sup_{1≤j≤n} |e(j)|^{2r+s} + 1] ((2r + s)|J|)^{2r+s} ∑_{\begin{matrix}|U|≤2r+s−1 \\ (V_{1},...,V_{p})∈U_{U} \end{matrix}} |H_{U,J}^{(n)}(V_{1}, …, V_{p})|

and

Σ_{2} = ∑_{(V_{1},...,V_{p})∈V} |H_{V,J}^{(n)}(V_{1}, …, V_{p})| ,

where V = {1, …, 2r + s}, and V denotes the set of those partitions of V whose elements are unions of the sets {1, 2}, …, {2r −1, 2r}, {2r +1}, …, {2r +s} and it contains at least one set such that it has a proper subset of the form {2j −1, 2j}, 1 ≤ j ≤ r. Here Σ_{1} bounds the contribution of those products e(j_{1}) ⋯ e(j_{2r+s}) in (4.15) which contain a term e(j_{l}) with j_{l} ∈ J, and Σ_{2} bounds the contribution of those products for which e(j_{l}) ∈ {1, …, n} \ J for all 1 ≤ l ≤ 2r + s, but do not appear in the expression defining H_{V,J}^{(n)}(r, s). The relations Σ_{1} ≤ const. n^{(2r+s−1)/2} and Σ_{2} ≤ const. n^{(2r+s−1)/2} hold because of formulas (4.10) and (4.11) respectively.

We shall prove by induction for s that

lim_{n→∞} n^{−s/2}H^{(n)}(0, s) = D_{s}  (4.16)

with the sequence D_{s} defined in (4.8). Indeed, (4.16) holds for s = 0 and s = 1, and for s ≥ 2 we can write

H^{(n)}(0, s) = n^{s/2}F_{n}^{s} − ∑_{(V_{1},...,V_{p})∈U_{s}\({1},...,{s})} H_{U,∅}^{(n)}(V_{1}, …, V_{p})

where U_{s} denotes the set of partitions of U = {1, …, s}. We get relation (4.16) by dividing in the last relation by n^{−s/2} and taking limit n → ∞ if we use relations (4.11), (4.14), the induction hypothesis, the relation lim_{n→∞} E_{n} = E, lim_{n→∞} F_{n} = F

and the fact that the set \{1, …, s\} contains \frac{s!}{2^{p}p!(s − 2p)!} partitions consisting of

p sets with cardinality 2 and s − 2p sets with cardinality 1, 1 < 2p ≤ s.

Clearly, for the expression G_{n} defined in (4.9′) G_{n}(j_{1}, …, j_{s}) = H_{J}(0, k − s) with J = {j_{1}, …, j_{s}}. Hence relations (4.16) and (4.14) imply that

lim_{n→∞} n^{−(k−s)/2}G_{n}(j_{1}, …, j_{s}) = D_{k−s}

and the convergence is uniform in (j_{1}, …, j_{s}). The last relation together with formula (4.9) imply Lemma 4. □

Proof of Theorem 4. We get by rewriting the expression (2.4) by means of the Hoeffding decomposition, and applying Lemma 4 that

n^{−k/2}U_{n} = V_{n} + η_{n}

with

V_{n} = ∑_{s=1}^{k} n^{−s/2}\binom{k}{s}\frac{1}{k!}D_{k−s} ∑_{\begin{matrix}1≤j_{p}≤n, for 1≤p≤s \\ j_{p}≠j_{p′} if p≠p′ \end{matrix}} e(j_{1}) ⋯ e(j_{s})f_{s}(X_{j_{1}}, …, X_{j_{s}})

and

η_{n} = ∑_{s=1}^{k} n^{−s/2} \frac{1}{k!} ∑_{\begin{matrix}1≤j_{p}≤n, for 1≤p≤s \\ j_{p}≠j_{p′} if p≠p′ \end{matrix}} ε_{n}^{(s)}(j_{1}, …, j_{s})f_{s}(X_{j_{1}}, …, X_{j_{s}}) .

The random variables f(X_{j_{1}}, …, X_{j_{s}}) and f(X_{j′_{1}}, …, X_{j′_{s}}) are uncorrelated if the sets {j_{1}, …, j_{s}} and {j′_{1}, …, j′_{s}} are different, since the functions f_{s} satisfy relation (1.4′). Hence formula (4.7′) implies that Eη_{n}^{2} → 0 as n → ∞, and n^{−1/2}U_{n} and V_{n} have the same limit distribution as n → ∞. By Theorem 3′ the random variables V_{n} have the limit distribution given in Theorem 4. □

Remark 2. If lim_{n→∞} F_{n} = ∞, and the remaining conditions of Theorem 4 hold and s is the smallest index such that the function f_{s} in (1.4) does not vanish identically, then the sequence n^{−k/2}F_{n}^{s−k}U_{n} converges in distribution to the stochastic integral

\frac{E^{s/2}}{s!(k − s)!} ∫ f_{s}(x_{1}, …, x_{s}) W ( dx_{1}) … W ( dx_{s})

as n → ∞. This can be proved similarly to Theorem 4, the only difference is that now the behavior of the coefficint B_{n} defined in (2.3) is different. In this case

B_{n}(j_{1}, …, j_{s}) ≈ n^{(k−s)/2}F_{n}^{k−s}e(j_{1}) ⋯ e(j_{s}) .

The situation is more complex in the case lim_{n→∞} F_{n} = 0. In this case some coefficients D_{j} may equal zero in the limit distribution appearing in Theorem 4. In particular, D_{j} = 0 if the index j is an odd number. Hence it may happen that the limit appearing in Theorem 4 equals zero, and we have to apply a different normalization to get a useful limit theorem. Here again a good asymptotics is needed for the function B_{n} in (2.3). In this case those great indices s count for which the function f_{s} does not vanish in the Hoeffding decomposition (1.4). But the situation is more complicated in this case. The asymptotic behavior of the

sums ∑_{j=1}^{n} e(j)^{r} can play role not only for r = 1 or 2. We omit a closer investigation

of this problem.

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