THE THEORY OF WIENER–ITÔ INTEGRALS IN VECTOR-VALUED GAUSSIAN STATIONARY RANDOM

FIELDS. PART II

PÉTER MAJOR

Abstract. This work is the continuation of my paper in Moscow Math. Journal Vol. 20, No. 4 in 2020. In that paper first I proved the existence of the spectral measure of a vector-valued stationary Gaussian random field. I also constructed there the vector-valued random spectral measure corresponding to this spectral measure, presented its most important properties, and defined the multiple Wiener–Itô integrals with respect to it. I also proved an important identity about the products of multiple Wiener–Itô integrals which is called the diagram formula. Here I prove with the help of these results the multivariate version of Itô’s formula which shows a relation between multiple Wiener–Itô integrals with respect to vector-valued random spectral measures and Wick polynomials. Wick polynomials are the multivariate versions of Hermite polynomials. I also prove a formula that expresses the shift transforms of a random variable given in the form of a multiple Wiener–Itô integral. These results enable us to rewrite certain non-linear functionals of a vector-valued stationary Gaussian random field in a useful form which suggests a limiting procedure that leads to interesting limit theorems. Finally, I show when this limiting procedure may be carried out, i.e., when the limit theorems suggested by our new representation of the investigated non-linear functionals are valid.

1. Introduction. The main results of the paper.

This work deals with the properties of vector-valued stationary Gaussian random fields. In particular, one of its subjects is the problem how to prove non-central limit theorems for certain non-linear functionals of such random fields. It is the continuation of paper [10]. These two papers deal with a generalized version of the problems studied in [8]. In Lecture Note [8] I investigated scalar-valued stationary random fields. I gave a good representation of non-linear functionals of a stationary Gaussian random field with the help of multiple Wiener–Itô integrals with respect to the random spectral measure of this Gaussian random field. The definition of random

2010 Mathematics Subject Classification. 60G10, 60G15, 60F99. Key words and phrases. Multiple Wiener–Itô integrals, multivariate version of Itô’s

formula, Wick polynomials, shift transformation, vague convergence of complex measures, non-central limit theorems.

spectral measures and multiple Wiener–Itô integrals with respect to it is explained in this Lecture Note. They made possible to prove some interesting non-central limit theorems.

This work together with paper [10] contain the natural multivariate version of the results in [8]. Their results provide a useful representation of non-linear functionals of a vector-valued stationary Gaussian random field with the help of (vector-valued) multiple Wiener–Itô integrals with respect to the random spectral measure of the underlying Gaussian field. The notions needed to understand the meaning of these results are also introduced in these works. Some useful properties of the multiple Wiener–Itô integrals are also proved. In a subsequent paper [11] I will prove the natural multivariate version of the non-central limit theorems formulated in [8] with the help of these results.

In Theorem 6 of his paper [1] Arcones formulated the multivariate version of the non-central limit theorems proved in [8]. Actually, this is the result we want to prove in [11]. But the formulation and proof of Arcones’ results are unsatisfactory. He assumed that the natural multivariate versions of some results in [8] hold true. But he did not formulate precisely the results he applied, and even the definitions of the notions needed in the formulation of his results are missing. In this work and in [10] I fill this gap.

Next, I briefly describe the content of the present work and compare it with the results of [8]. In this description I also mention some results in [10].

In paper [10] I have worked out the theory of vector-valued stationary Gaussian random fields. I introduced the spectral measure and the random spectral measure of a vector-valued stationary Gaussian random field. I also defined a class of generalized stationary Gaussian fields together with their spectral and random spectral measures. The subsequent results in [10] are valid both for classical and for these new generalized stationary Gaussian random fields. I defined multiple Wiener–Itô integrals with respect to random spectral measures and proved some important results about them. In particular, I proved the so-called diagram formula which enables us to rewrite the product of multiple Wiener–Itô integrals in the form of a sum of appropriately defined multiple Wiener–itô integrals. The work [8] contained the one-dimensional version of these results.

In [8] I considered a scalar-valued stationary Gaussian random field X(p), p ∈ ℤ^{ν}, on the ν-dimensional integer lattice of the Euclidean space ℝ^{ν}, with expectation EX(p) = 0. I have introduced the Hilbert space H consisting of those random variables with finite second moment which are measurable with respect to the σ-algebra generated by the random variables X(p), p ∈ ℤ^{ν}, of our random field. Here the usual scalar product ⟨ξ, η⟩ = Eξη is applied, and our goal is to understand the behavior of this Hilbert space.

WIENER–ITÔ INTEGRALS IN GAUSSIAN STATIONARY RANDOM FIELDS II 3

It is proved in [8] that this Hilbert space H has a natural representation as the direct sum H = H_{0} + H_{1} + H_{2} + ⋯ of orthogonal subspaces H_{n}, 0 ≤ n < ∞, which are invariant subspaces of the shift transforms in the underlying stationary Gaussian random field, and the subspace H_{n} consists of those random variables which can be written in the form of an n-fold Wiener–Itô integral with respect to the random spectral measure of the underlying Gaussian random field. (Actually, [8] gives a more detailed description of the structure of the space H with the help of the so-called Fock space representation. The definition of the Fock space is given on page 28 of [8], and it is denoted by Exp H_{G} there.) The relation between the Fock space and the Hilbert space H is formulated in Theorem 4.2 of [8] with the help of this notation. Here I do not discuss this result.) The proof of the above representation of the Hilbert space H is based on the so-called Itô formula which shows a relation between Hermite polynomials and multiple Wiener–Itô integrals. Another important result in [8], whose proof is also based on Itô’s formula is a useful formula that enables us to calculate the shift transforms of a random variable presented in the form of a multiple Wiener–Itô integral.

Section 2 of this paper deals with the generalization of the above results if we are working with vector-valued stationary Gaussian random fields. First, the formulation and proof of the multivariate version of Itô’s formula are given. In this result, the Wick polynomials, the multivariate generalizations of Hermite polynomials, take the role of Hermite polynomials. At the start of Section 2, I recall their definition and most important properties from [8]. Then I prove the multivariate version of Itô’s formula by adapting the method of proof of its one-dimensional version. I discuss the proof of this result in Appendix A of this paper. I explain there the picture behind the definition of Wick polynomials, about the idea of the proof of the Itô’s formula, and why Wick polynomials appear in Itô’s formula when vectorvalued Gaussian random fields are considered. I also discuss the question how the remaining results of Section 2 are proved with the help of Itô’s formula.

In this paper I consider vector-valued stationary Gaussian random fields X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν}, with expectation EX(p) = 0, p ∈ ℤ^{ν}, and a Hilbert space H defined similarly in the case of scalar-valued random fields. It is the Hilbert space consisting of those random variables with finite second moment which are measurable with respect to the σalgebra generated by the random vectors X(p), p ∈ ℤ^{ν}, of our random field. Similarly to the one-dimensional case, there is a decomposition of the Hilbert space H to the direct product H = H_{0} + H_{1} + H_{2} + ⋯ of orthogonal subspaces, H_{n}, 0 ≤ n < ∞, which are invariant subspaces of the shift transformations in the underlying stationary random field. But in the case of vector-valued stationary Gaussian random fields we can prove only a weaker result about the behavior of the subspaces H_{n} than in the scalar-valued case. It is proved in Proposition 2.3 that the elements of an everywhere dense linear subspace of H_{n} can be written in the form of a finite sum of multiple Wiener–Itô integrals of order n. More explicitly, the Wick polynomials of order n can be written as a finite sum of multiple Wiener–Itô integrals of order n, and they constitute an everywhere dense linear subspace of H_{n}. On the other hand, we cannot represent all elements of H_{n} in such a way. But even this weaker result is sufficient for our purposes.

The last result of Section 2, Proposition 2.4, contains a useful formula about the calculation of the shift transforms of a random variable given in the form of a multiple Wiener–Itô integral. This formula is similar to the analogous result in the case of scalar-valued stationary random fields.

In Section 3 we deal with the question how the previously proved results can be applied in the investigation of limit theorems for non-linear functionals of vector-valued stationary Gaussian random fields.

In several interesting cases (and the problem investigated in paper [11] belongs to them) the limit problem we are interested in can be reformulated with the help of Itô’s formula and an appropriate rescaling of certain multiple Wiener–Itô integrals we are working with to the study of a sequence of random variables presented in a very special form, and in Section 3 we are investigating limit theorems for such sequences of random variables.

We consider a sequence of random variables Z_{N}, N = 1, 2, …, presented as a finite sum of k-fold Wiener–Itô integrals with respect to a ddimensional random spectral measure. We are interested in the behavior of such a sequeqnce of random variables Z_{N}, N = 1, 2, …, whose elements are defined by formula (3.9) with the help of random spectral measures Z_{G^{(N)}} = (Z_{G^{(N)},1}, …, Z_{G^{(N)},d}) which correspond to some spectral mea-

sures G^{(N)} = (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, defined on some torus [−A_{N}, A_{N})^{ν}, where A_{N} → ∞ as N → ∞, and we integrate some kernel functions h_{j_{1},...,j_{n}}^{(N)}(x_{1}, …, x_{d}) with respect to them. Let us remark that the random variables Z_{N} introduced in (3.9) are sums of finitely many multiple Wiener– Itô integrals. Each element of this sum is indexed by some vector (j_{1}, …, j_{k}), and the set of these indices does not depend on the parameter N. We expect that if both the spectral measures G^{(N)} and the kernel functions h_{(j_{1},...,j_{n})}^{N} appearing in the definition of the random variables Z_{N} behave nicely, then these random variables have a limit as N → ∞. Somewhat more explicitly, we expect that if the spectral measures G^{(N)} converge to a spectral measure G^{(0)} of a d-dimensional (generalized) stationary Gaussian random field, and the kernel functions h_{(j_{1},...,j_{k})}^{N}(x_{1}, …, x_{d}) converge to some nice functions h_{(j_{1},...,j_{k})}^{0}(x_{1}, …, x_{d}) as N → ∞, then the random variables Z_{N} converge in distribution to the random variable Z_{0} defined in (3.12) with the help of multiple Wiener–Itô integrals of the kernel functions h_{(j_{1},...,j_{k})}^{0}(x_{1}, …, x_{d}) with respect to the random spectral measure Z_{G^{(0)}} corresponding to the spectral measure G^{(0)}. Naturally, in the formulation of such a result we have to clarify what kind of limit should hold for the spectral measures G^{(N)} and for the kernel functions h_{(j_{1},...,j_{k})}^{N}(x_{1}, …, x_{d}).

The main result of Section 3 is Proposition 3.1. First the random variables Z_{N}, N = 0, 1, 2, …, mentioned in the above discussion are precisely defined in formulas (3.9) and (3.12), and then Proposition 3.1 yields a sufficient condition for the weak convergence of the random variables Z_{N} to Z_{0} as N → ∞. This condition consists of two parts. The first part of them demands that the spectral measures G^{(N)} converge to the spectral measure G^{(0)} and the kernel functions h_{(j_{1},...,j_{k})}^{N}(x_{1}, …, x_{d}) converge to h_{(j_{1},...,j_{k})}^{0}(x_{1}, …, x_{d}) as N → ∞ in a right way. This is the condition suggested by the heurisitic argument of the previous paragraph. But to satisfy the desired weak convergence we have to impose another condition formulated in condition (b) of Proposition 3.1. This is a compactness type condition. Heuristically, it has the following content. There is such a compact set in ℝ^{kν} that all multiple Wiener–Itô integrals appearing in the definition of one of the random variables Z_{N}, N = 1, 2, … are essentially concentrated in this compact set. In the Appendix I explain the role of this condition with the help of some examples.

The condition about the limiting behavior of the kernel functions h_{(j_{1},...,j_{k})}^{N}(x_{1}, …, x_{d}) is formulated in condition (a) of Proposition 3.1. The condition imposed on the convergence of the spectral measures G^{(N)} deserves special attention.

We are working with spectral measures (G^{(N)}) = (G_{j,j′}^{(N)}), N = 0, 1, 2, …, 1 ≤ j, j′ ≤ d, and we impose such a condition that for all indices 1 ≤ j, j′ ≤ d the coordinates G_{j,j′}^{(N)} of our spectral measures converge to the

coordinate G_{j,j′}^{(0)} as N → ∞ in an appropriate way. We want to consider such cases when G^{(0)} is the spectral measure of such a generalized stationary Gaussian random field which belongs to the class of generalized stationary Gaussian random fields constructed in Section 4 of [10]. This must be taken into consideration when we want to decide what kind of convergence of the spectal measures we impose in Proposition 3.1. The coordinates of the spectral measures of the generalized stationary Gaussian random fields we are working with are complex measures on ℝ^{ν} with locally finite total variation. (The definition of this notion is explained in Section 4 of [10]). In Section 3 of this paper we define the vague convergence of complex measures on ℝ^{ν} with locally finite total variation, and in Proposition 3.1 we impose this convergence for the coordinates G_{j,j′}^{(N)} of the sequence spectral measures G^{(N)}.

In Lemma 8.3 of [8] the scalar-valued version of Proposition 3.1 is formulated. In that result, the notion of vague convergence also appears. But the definition of the vague convergence in [8] and in this paper is slightly different. In [8] the vague convergence of locally finite measures, while here the vague convergence of complex measures on ℝ^{ν} with locally finite total variation is defined. The difference between these two definitions is greater than it may seem at the first sight. Let me remark that while the restriction of a complex measure on ℝ^{ν} with locally finite total variation to the measurable subsets of a compact subset of ℝ^{ν} is a complex measure, there are such complex measures on ℝ^{ν} with locally finite total variation which are not complex measures on the measurable subsets of ℝ^{ν}. Hence we are considering the vague convergence of different objects in this paper and in [8].

As I mentioned before, Proposition 3.1 is a multivariate generalization of Lemma 8.3 in [8]. Its proof applies the ideas of the proof of this lemma, but some additional difficulties also had to be overcome.

Section 3 contains still another result. It is Lemma 3.2 which may be useful when we apply Proposition 3.1. It states that if all coordinates G_{j,j′}^{(N)} of

a sequence of spectral measures G^{(N)} = (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, N = 1, 2, …,

converge vaguely to some complex measures G_{j,j′}^{(0)} with finite total variation,

then also G^{(0)} = (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, is a positive semidefinite matrix valued even measure on ℝ^{ν}.

This result is useful because it helps us to decide when the limit matrix G^{(0)} = (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, is the spectral measure of a (generalized) stationary Gaussian random field. (See the remark after the formulation of Lemma 3.2).

In this paper together with [10] and also in the work [8] I applied a version of the multiple Wiener–Itô integrals introduced by Itô in his paper [7]. (Itô called these random integrals multiple Wiener integrals in his paper.) I applied, instead of the random integral introduced by Itô a version of it introduced by Dobrushin in [4], and considered its generalization when vector-valued stationary Gaussian random fields are taken. At the end of the Introduction, I give a short explanation why it was useful to work with this version of the random integrals introduced by Itô in [7], and I also briefly mention another example where a multidimensional time generalization of the Itô integral was introduced in order to study some problems.

Itô considered in his paper [7] a measure space (T, B, m), and a set of (jointly) Gaussian random variables β(E) indexed by such sets E ∈ B for which m(E) < ∞, and the joint distibution of these random variables is determined by the relations Eβ(E) = 0, and Eβ(E)β(E′) = m(E ∩ E′). He also imposed the following continuity property for the measure m. For any set E ∈ B with m(E) < ∞ and ε > 0 there exists a finite decomposition E = ∑_{i=1}^{n} E_{i} such that m(E_{i}) < ε for all i = 1, …, n. Itô defined the p-fold Wiener integrals of square integrable functions (with respect to the product measure m^{p}) with respect to the random measure β(E, ω) for all p = 0, 1, … In the definition of the random measure β(E, ω) the parameter set consists of those measurable sets E for which m(E) < ∞. Then he gave a useful representation of all square integrable random variables measurable with respect to the σ-algebra generated by the random variables β(E) as a sum of multiple Wiener integrals with different multiplicity.

To present such a representation Itô proved some useful results about the properties of multiple Wiener integrals. In particular, he proved an identity that was later called Itô’s formula. He showed with its help an important relation between multiple Wiener integrals and Hermite polynomials.

In paper [4] Dobrushin proved for the sake of investigation of non-linear functionals of stationary Gaussian random fields such a version of the random integrals in Itô’s paper [7], where he worked in the Euclidean space (ℝ^{ν}, B), and he replaced the random measure β(E, ω) by the random spectral measure of a stationary Gaussian random field. I described these results in more detail in [8], and this paper together with [10] yield a generalization of these results when vector-valued stationary Gaussian random fields are considered. These works show that results similar to those of the paper [4] hold if we work with random spectral measures instead of the random measure β(E, ω) applied in [4]. One may ask why it was useful to work with random spectral measures instead of the application of the results in [4].

I want to show that by working with random spectral measures we could prove some Fourier analysis type results which were useful in our investigation. Proposition 2.4 of this paper is such a result. Here we calculate the shift transform T_{u} of a random variable presented in the form of a multiple Wiener–Itô integral. The shifted random variable is presented in a Fourier transform type form. Indeed, Proposition 2.4 says that the shift transform T_{u}Y of a random variable Y given in formula (2.6) equals the multiple Wiener–Itô integral given in formula (2.8). This result together with Itô’s formula which enables us very often to rewrite the random variables we are working with as sums of multiple Wiener–Itô integrals help in the study of limit theorems. The discussion at the beginning of Section 3 shows an example of how to apply these results.

The application of multiple stochastic integrals turned out to be useful also in the investigation of some other problems. I mention only very briefly the contribution of Wong–Zakai–Yor to the theory of multiple stochastic integrals as an example for it. They discussed the following problem. The study of Itô integrals is very closely related to the study of martingales. For example, any continuous-time square integrable martingale with continuous trajectories adapted to a Wiener process has a canonical representation as an Itô integral with respect to the underlying Wiener process. The abovementioned mathematicians were looking for the multidimensional time version of this result. Naturally, to formulate this result first the multidimensional time Wiener processes and martingales must be defined. But these objects are defined in the literature. I would mention that in particular, the definition of the multidimensional time martingales demands special attention.

A multidimensional version of the result about the canonical representation of square-integrable martingales can be proved. But in that representation, not only the Itô integrals (integrals with multidimensional time which also have to be defined) but also multiple Itô integrals appear. The precise formulation of this result would demand a very long explanation, hence I omit it. The interested reader can find it in paper [6] and in its list of references. This work also discusses with the help of this result a statistical problem, where the main point of the solution is the calculation of a Radon–Nikodym derivative.

2. Wick polynomials and their relation to multipe Wiener–Itô integrals

In the case of scalar-valued stationary Gaussian random fields (i.e., if d = 1) there is a so-called Itô formula (see Theorem 4.3 in [8]) which shows an important relation between multiple Wiener–Itô integrals and Hermite polynomials. Here I present its multivariate version, where Wick polynomials take the role of the Hermite polynomials. Wick polynomials are the natural multi-dimensional generalizations of Hermite polynomials. I shall also discuss an important consequence of the multivariate version of the Itô formula. This formula enables us to present a large class of random variables in the form of a sum of multiple Wiener–Itô integrals. Besides, there is a useful formula for the calculation of the shift transforms of such random variables which are given in the form of a sum of multiple Wiener–Itô integrals. As we shall see, this formula is very useful in the study of limit theorems for non-linear functionals of a vector-valued stationary Gaussian field. I shall explain in the first part of the Appendix the relation between the proof of Itô’s formula in the scalar and in the vector-valued case. In that explanation I also write about the definition of the Wick polynomials and their role in the proof.

First I recall the definition of Wick polynomials and some results about their most important properties. Here I follow the discussion in Section 2 of [8].

Let X_{t}, t ∈ T, be a set of jointly Gaussian random variables indexed by a parameter set T, and such that EX_{t} = 0 for all t ∈ T. We define the following real Hilbert spaces H_{1} and H. A square integrable (real valued) random variable is in H if and only if it is measurable with respect to the σ-algebra B = B(X_{t}, t ∈ T), and the scalar product in H is defined as ⟨ξ, η⟩ = Eξη, ξ, η ∈ H. The Hilbert space H_{1} ⊂ H is the subspace of H generated by the finite linear combinations ∑ c_{j}X_{t_{j}}, t_{j} ∈ T, with real coefficients. We consider only such sets of Gaussian random variables X_{t} for which H_{1} is separable. Otherwise X_{t}, t ∈ T, can be arbitrary, but the most interesting case for us is when T = ℤ^{ν} × {1, …, d}, and the original Gaussian random variables we are working with are the coordinates X_{j}(p), j ∈ {1, …, d}, p ∈ ℤ^{ν}, of a vector-valued stationary Gaussian random field X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν}.

To define the Wick polynomials and to get their most important properties we need the following result formulated in Theorem 2.1 of [8].

Theorem 2A. Let Y_{1}, Y_{2}, … be an orthonormal basis in the Hilbert space H_{1} defined above with the help of a set of Gaussian random variables X_{t}, t ∈ T. Then the set of all possible finite products H_{j_{1}}(Y_{l_{1}}) ⋯ H_{j_{k}}(Y_{l_{k}}) is a complete orthogonal system in the Hilbert space H defined above. (Here, and in the subsequent discussion H_{j}(·) denotes the j-th Hermite polynomial with leading coefficient 1.)

Let H_{≤n} ⊂ H, n = 1, 2, …, (with the previously introduced Hilbert space H) denote the linear subspace of the Hilbert space H which is the closure of the linear space consisting of the elements P_{n}(X_{t_{1}}, …, X_{t_{m}}), where P_{n} runs through all polynomials of degree less than or equal to n, and the integer m and indices t_{1}, …, t_{m} ∈ T are arbitrary. Let H_{0} = H_{≤0} consist of the constant functions, and let H_{n} = H_{≤n} ⊖ H_{≤n−1}, n = 1, 2, …, where ⊖ denotes orthogonal completion. It is clear that the Hilbert space H_{1} given in this definition agrees with the previously defined Hilbert space H_{1}. If ξ_{1}, …, ξ_{m} ∈ H_{1}, and P_{n}(x_{1}, …, x_{m}) is a polynomial of degree n, then P_{n}(ξ_{1}, …, ξ_{m}) ∈ H_{≤n}. Then Theorem 2A implies that

H = H_{0} + H_{1} + H_{2} + ⋯,  (2.1)

where + denotes direct sum. Now I present the definition of Wick polynomials.

Definition of Wick polynomials. Let P (x_{1}, …, x_{m}) be a homogeneous polynomial of degree n, and let a set of (jointly Gaussian) random variables ξ_{1}, …, ξ_{m} ∈ H_{1} be given. The Wick polynomial : P (ξ_{1}, …, ξ_{m}): determined by them is the orthogonal projection of the random variable P (ξ_{1}, …, ξ_{m}) to the above-defined subspace H_{n} of the Hilbert space H. The Wick polynomial of a homogeneous polynomial of degree n will be called a Wick polynomial of order n.

In the sequel we shall use the notation : P (ξ_{1}, …, ξ_{m}): for the Wick polynomial corresponding to a homogeneous polynomial P (x_{1}, …, x_{m}) with arguments ξ_{1}, …, ξ_{m}, ξ_{j} ∈ H_{1} for all 1 ≤ j ≤ m. It may happen that a random variable ζ can be expressed in two different forms as a homogeneous polynomial of some random variables from H_{1}, i.e., ζ = P_{1}(ξ_{1}, …, ξ_{m}), and ζ = P_{2}(ξ_{1}, …, ξ_{m}), and P_{1} ≠ P_{2}. But in such a case

: P_{1}(ξ_{1}, …, ξ_{m}): =: P_{2}(ξ_{1}, …, ξ_{m}): ,

i.e., the value of a Wick polynomial : P (ξ_{1}, …, ξ_{m}): does not depend on the representation of the random variable P (ξ_{1}, …, ξ_{m}).

It is clear that Wick polynomials of different degree are orthogonal. Given some ξ_{1}, …, ξ_{m} ∈ H_{1} define the subspaces H_{≤n}(ξ_{1}, …, ξ_{m}) ⊂ H_{≤n}, n = 1, 2, …, as the set of all polynomials of the random variables ξ_{1}, …, ξ_{m} with degree less than or equal to n. Let H_{≤0}(ξ_{1}, …, ξ_{m}) = H_{0}(ξ_{1}, …, ξ_{m}) = H_{0}, and H_{n}(ξ_{1}, …, ξ_{m}) = H_{≤n}(ξ_{1}, …, ξ_{m}) ⊖ H_{≤n−1}(ξ_{1}, …, ξ_{m}). With the help of this notation I formulate the following result given in Proposition 2.2 of [8].

Theorem 2B. Let P (x_{1}, …, x_{m}) be a homogeneous polynomial of degree n. Then : P (ξ_{1}, …, ξ_{m}): equals the orthogonal projection of P (ξ_{1}, …, ξ_{m}) to H_{n}(ξ_{1}, …, ξ_{m}).

This result has the following important consequences formulated in Corollaries 2.3 and 2.4 in [8].

Corollary 2C. Let ξ_{1}, …, ξ_{m} be an orthonormal system in H_{1}, and let

P (x_{1}, …, x_{m}) = ∑ c_{j_{1},...,j_{m}}x^{j_{1}} ⋯ x_{m}^{j_{m}}

be a homogeneous polynomial, i.e., let j_{1} + ⋯ + j_{m} = n with some fixed number n for all sets (j_{1}, …, j_{m}) appearing in this summation. Then

: P (ξ_{1}, …, ξ_{m}): = ∑ c_{j_{1},...,j_{m}}H_{j_{1}}(ξ_{1}) ⋯ H_{j_{m}}(ξ_{m}).

In particular,

: ξ^{n}: = H_{n}(ξ) if ξ ∈ H_{1}, and Eξ^{2} = 1.

Corollary 2D. Let ξ_{1}, ξ_{2}, … be an orthonormal basis in H_{1}. Then the random variables H_{j_{1}}(ξ_{1}) ⋯ H_{j_{k}}(ξ_{k}), k = 1, 2, …, j_{1} + ⋯ + j_{k} = n, form a complete orthogonal basis in H_{n}.

In the proof of the Itô formula for scalar-valued stationary random fields we needed, besides the diagram formula, the following important recursive formula for Hermite polynomials which is contained for example in Lemma 5.2 of [8].

H_{n}(x) = xH_{n−1}(x) − (n − 1)H_{n−2}(x) for n = 1, 2, …,  (2.2)

with the notation H_{−1}(x) ≡ 0 in the case n = 1.

In the next result I formulate a multivariate version of this formula for Wick polynomials.

Proposition 2.1. Let U_{1}, …, U_{n+1}, n ≥ 1, be elements in H_{1}. Then

: U_{1} ⋯ U_{n}: U_{n+1}  (2.3)

=: U_{1} ⋯ U_{n}U_{n+1} : + ∑_{s=1}^{n}: U_{1} ⋯ U_{s−1}U_{s+1} ⋯ U_{n}: EU_{s}U_{n+1}.

In the special case n = 1 this formula is meant as U_{1}U_{2} =: U_{1}U_{2} : + EU_{1}U_{2}.

Proof of Proposition 2.1. Formula (2.3) clearly holds if all random variables U_{j}, 1 ≤ j ≤ n + 1 agree, and EU_{1}^{2} = 1, since in this case the left-hand side of (2.3) equals U_{1}H_{n}(U_{1}), while its right-hand side equals H_{n+1}(U_{1}) + nH_{n−1}(U_{1}) by Corollary 2C, and these two expressions are equal by formula (2.2). A somewhat more complicated, but similar argument shows that this formula also holds if the sequence U_{1}, …, U_{n} consists of some independent random variables V_{1} …, V_{k} with standard normal distribution, the random variable V_{p} is contained in the sequence U_{1},…, U_{n} with multiplicity l_{p}, 1 ≤ p ≤ k, and finally U_{n+1} is either one of these random variables V_{p}, 1 ≤ p ≤ k, or it is a random variable V_{k+1} with standard normal distribution which is independent of all of them.

Indeed, if U_{n+1} = V_{p} with some 1 ≤ p ≤ k, then the left-hand side of (2.3) equals

H_{l_{1}}(V_{1}) ⋯ H_{l_{k}}(V_{k})V_{p},

while the right-hand side equals

H_{l_{1}}(V_{1}) ⋯ H_{l_{p−1}}(V_{p−1})H_{l_{p}+1}(V_{p})H_{l_{p+1}}(V_{p+1}) ⋯ H_{l_{k}}(V_{k})

+l_{p} H_{l_{1}}(V_{1}) ⋯ H_{l_{p−1}}(V_{p−1})H_{l_{p}−1}(V_{p})H_{l_{p+1}}(V_{p+1}) ⋯ H_{l_{k}}(V_{k})

by Corollary 2C. A comparison of these expressions together with relation (2.2) imply that identity (2.3) holds in this case. If U_{n+1} = V_{k+1}, then the left-hand side of (2.3) equals

H_{l_{1}}(V_{1}) ⋯ H_{l_{k}}(V_{k})V_{k+1},

and the right-hand side also equals H_{l_{1}}(V_{1}) ⋯ H_{l_{k}}(V_{k})V_{k+1}. Hence formula (2.3) holds in this case, too.

In the general case we can choose some independent Gaussian random variables Z_{1}, …, Z_{m} in H_{1} with variance 1 in such a way that our random variables U_{1}, …, U_{n+1} can be expressed as their linear combination, i.e., U_{p} = ∑_{l=1}^{m} c_{p,l}Z_{l} with some coefficients c_{l,m}. We have already seen that formula (2.3) is valid in the special case when all random variables U_{p} equal one of the random variables Z_{j}, i.e., if U_{p} = Z_{j(p)} with some 1 ≤ j(p) ≤ m for all 1 ≤ p ≤ n + 1. Since the expressions of both sides of (2.3) are multi-linear functionals on the n-fold direct product H_{1} × ⋯ × H_{1}, this implies that formula (2.3) also holds for the random variables U_{1}, …, U_{n+1}. Proposition 2.1 is proved. □

We can prove the multivariate version of Itô’s formula with the help of Proposition 2.1 and the diagram formula for multiple Wiener–Itô integrals for vector-valued stationary Gaussian random fields formulated in Section 6 of [10].

Before its formulation, I make a remark about the notation in this section. In the formulation of Itô’s formula the notation K_{1,j} appears. This no-

tion was introduced in Lemma 3.2 of [10]. It is a real Hilbert space, and it contains those functions u ∈ K_{1,j} for which we defined the random integral ∫ u(x)Z_{G,j}( dx) with respect to the j-th coordinate Z_{G,j} of the random spectral measure Z_{G} = (Z_{G,1}, …, Z_{G,d}), and the value of this integral is a real valued random variable. In Section 4 of [10] this notion is defined also in the case of generalized random spectral measure. Later, at the beginning of Section 5 of [10] its multidimensional generalization, the real Hilbert space K_{n,j_{1},...,j_{n}} = K_{n,j_{1},...,j_{n}}(G_{j_{1},j_{1}}…. , G_{j_{n},j_{n}}) is defined for all n = 1, 2, . . .. It consists of those functions f(x_{1}, …, x_{n}) for which the n-fold (real valued) Wiener–Itô integral

I_{n}(f|j_{1}, …, j_{n}) = ∫ f(x_{1}, …, x_{n})Z_{G,j_{1}}( dx_{1}) … Z_{G,j_{n}}(d x_{n})

is defined. The Hilbert space K_{1,j} is a special case of these Hilbert spaces with n = 1. The Hilbert spaces K_{n,j_{1},...,j_{n}} will appear also in this work. At some points I shall also work with the class of simple functions K̂_{n,j_{1},...,j_{n}} ⊂ K_{n,j_{1},...,j_{n}} defined also in Section 5 of [10]. The multiple Wiener–Itô integrals were first defined for simple functions which are adapted to some regular system (defined also in Section 5 of [10]), and the multiple Wiener–Itô integrals were defined in the general case by means of a good approximation of the functions f ∈ K_{n,j_{1},...,j_{n}} by simple functions f ∈ K̂_{n,j_{1},...,j_{n}}.

Theorem 2.2. Multivariate version of Itô’s formula. Let us have some vector-valued stationary Gaussian random field with a vector-valued random spectral measure Z_{G} = (Z_{G,1}, …, Z_{G,d}). Let us consider some functions φ_{p} ∈ K_{1,j_{p}}, 1 ≤ p ≤ n, 1 ≤ j_{p} ≤ d, and define with their help the random variables U_{p} = ∫ φ_{p}(x)Z_{G,j_{p}}( dx) ∈ H_{1}, 1 ≤ p ≤ n. The identity

: U_{1} ⋯ U_{n}:  (2.4)
= ∫ φ_{1}(x_{1})φ_{2}(x_{2}) ⋯ φ_{n}(x_{n})Z_{G,j_{1}}( dx_{1})Z_{G,j_{2}}( dx_{2}) ⋯ Z_{G,j_{n}}( dx_{n})

holds.

Proof of Theorem 2.2. Relation (2.4) clearly holds for n = 1. We prove by induction that it holds for n + 1 if it holds for k ≤ n. In the proof we apply the Corollary of Theorem 6.1 from [10], (i.e., the corollary of the diagram formula from that paper) with the choice

h_{1}(x_{1}, …, x_{n}) = φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})
h_{2}(x) = φ_{n+1}(x),

and the random spectral measure Z_{G,j′_{1}} is chosen as Z_{G,j′_{1}} = Z_{G,j_{n+1}}, where Z_{G,j_{n+1}} is the random spectral measure appearing in the definition of U_{n+1}. We can write with this choice the identity

∫ φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})Z_{G,j_{1}}( dx_{1}) ⋯ Z_{G,j_{n}}( dx_{n}) ∫ φ_{n+1}(x)Z_{G,j_{n+1}}( dx) (2.5)
= ∫ φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})φ_{n+1}(x_{n+1})Z_{G,j_{1}}( dx_{1}) ⋯ Z_{G,j_{n+1}}( dx_{n+1})

+ ∑_{p=1}^{n} EU_{p}U_{n+1} ∫ φ_{1}(x_{1}) ⋯ φ_{p−1}(x_{p−1})φ_{p+1}(x_{p}) ⋯ φ_{n}(x_{n−1})

Z_{G,j_{1}}( dx_{1}) ⋯ Z_{G,j_{p−1}}( dx_{p−1})Z_{G,j_{p+1}}( dx_{p}) ⋯ Z_{G,j_{n}}( dx_{n−1}),

since formula (6.19) in [10] gives this identity with our choice of h_{1} and h_{2}. To see this observe that with these functions h_{1} and h_{2} the function h_{γ_{p}} in the formulation of the corollary of Theorem 6.1 in [10] equals for p ≠ 0

h_{γ_{p}}(x_{1}, …, x_{n−1}) = φ_{1}(x_{1}) ⋯ φ_{p−1}(x_{p−1})φ_{p+1}(x_{p}) ⋯ φ_{n}(x_{n−1})
∫ φ_{p}(x_{n})\overline{φ_{n+1}(x_{n})}G_{j_{p},j_{n+1}}( dx_{n})
= φ_{1}(x_{1}) ⋯ φ_{p−1}(x_{p−1})φ_{p+1}(x_{p}) ⋯ φ_{n}(x_{n−1})EU_{p}U_{n+1}

since U_{n+1} = \overline{U_{n+1}}, and by formula (3.7) in [10] which expresses the scalar product of two one-fold random integrals

EU_{p}U_{n+1} = EU_{p}\overline{U_{n+1}} = E (∫ φ_{p}(x)Z_{G,j_{p}}( dx)\overline{∫ φ_{n+1}(x)Z_{G,j_{n+1}}( dx)})
= ∫ φ_{p}(x_{n})\overline{φ_{n+1}(x_{n})}G_{j_{p},j_{n+1}}( dx_{n}),

and for p = 0

h_{γ_{0}}(x_{1}, …, x_{n+1}) = φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})φ_{n+1}(x_{n+1}).

Corollary of Theorem 6.1 in [10] with the above form of the kernel functions h_{γ_{p}}, 0 ≤ p ≤ n, in it imply formula (2.5). Formula (2.5) together with our induction hypothesis imply that

∫ φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})φ_{n+1}(x_{n+1})Z_{G,j_{1}}( dx_{1}) ⋯ Z_{G,j_{n+1}}( dx_{n+1})

=: U_{1} ⋯ U_{n}: U_{n+1} − ∑_{p=1}^{n}: U_{1} ⋯ U_{p−1}U_{p+1} ⋯ U_{n}: EU_{p}U_{n+1}.

In the case n = 1 this formula means that

∫ φ_{1}(x_{1})φ_{2}(x_{2})Z_{G,j_{1}}( dx_{1})Z_{G,j_{2}}( dx_{2}) = U_{1}U_{2} − EU_{1}U_{2}.

By comparing the last formula with (2.3) we get that the statement of Theorem 2.2 holds also for n + 1. Theorem 2.2 is proved. □

In Theorem 2.2 we rewrote some Wick polynomials of special form as multiple Wiener–Itô integrals. This enables us to express a sum of such Wick polynomials as the sum of multiple Wiener–Itô integrals. This implies that all Wick polynomials of random variables from some H_{1,j}, 1 ≤ j ≤ d, can be written in the form of a sum of multiple Wiener–Itô integrals. In the next simple corollary of Theorem 2.2 I describe this result in a more explicit form.

To formulate this result let us introduce the following notation. Let us fix some numbers n ≥ 1 (the order of the homogeneous polynomial we are considering), m ≥ 1 and some functions φ_{j,k}(x) ∈ K_{1,j}, 1 ≤ j ≤ d, 1 ≤ k ≤ m, and define the random variables

ξ_{j,k} = ∫ φ_{j,k}(x)Z_{G,j}( dx), 1 ≤ j ≤ d, 1 ≤ k ≤ m.

Then ξ_{j,k} ∈ H_{1,j}. (We defined the real Hilbert space H_{1,j} in the formulation of Lemma 3.2 of [10]. This Lemma 3.2 stated that the elements of H_{1,j} can be given in the form of the above integral.)

In the next corollary, we consider homogeneous polynomials of these random variables ξ_{j,k}, and express the Wick polynomials corresponding to them in the form of a sum of multiple Wiener–Itô integrals.

Corollary of Theorem 2.2. Let us consider a homogeneous polynomial

P (x_{j_{s},k_{s}}, 1 ≤ j_{s} ≤ d, 1 ≤ k_{s} ≤ m for all 1 ≤ s ≤ n)

= ∑_{1≤j_{s}≤d for all 1≤s≤n 1≤k_{s}≤m for all 1≤s≤n} a_{j_{1},k_{1},...j_{n},k_{n}}x_{j_{1},k_{1}}x_{j_{2},k_{2}} ⋯ x_{j_{n},k_{n}}

of order n of the variables x_{j_{s},k_{s}} with indices 1 ≤ j_{s} ≤ d and 1 ≤ k_{s} ≤ m for all 1 ≤ s ≤ n and real coefficients a_{j_{1},k_{1},...,j_{n},k_{n}}.

If we replace the variables x_{j_{s},k_{s}} with the random variables

ξ_{j_{s},k_{s}} = ∫ φ_{j_{s},k_{s}}(x)Z_{G,j_{s}}( dx)

in this polynomial (we choose a function φ_{j,k} ∈ K_{1,j} in the definition of ξ_{j,k}), then we get the following homogeneous polynomial of some jointly Gaussian random variables.

P (ξ_{j_{s},k_{s}}, 1 ≤ j_{s} ≤ d, 1 ≤ k_{s} ≤ m for all 1 ≤ s ≤ n)

= ∑_{1≤j_{s}≤d for all 1≤s≤n 1≤k_{s}≤m for all 1≤s≤n} a_{j_{1},k_{1},...j_{n},k_{n}}ξ_{j_{1},k_{1}}ξ_{j_{2},k_{2}} ⋯ ξ_{j_{n},k_{n}}.

With the help of this expression we can define the Wick polynomial

: P (ξ_{j_{s},k_{s}}, 1 ≤ j_{s} ≤ d, 1 ≤ k_{s} ≤ m for all 1 ≤ s ≤ n): .

This Wick polynomial can be expressed as a sum of multiple Wiener–Itô integrals in the following way.

Let us consider for all sequences of indices {(j_{s}, k_{s}), : 1 ≤ s ≤ n} with 1 ≤ j_{s} ≤ d, 1 ≤ k_{s} ≤ d for all 1 ≤ s ≤ n the function

f_{j_{1},k_{1},...,j_{n},k_{n}}(x_{1}, …, x_{n}) = φ_{j_{1},k_{1}}(x_{1}) ⋯ φ_{j_{n},k_{n}}(x_{n}) ∈ K_{n,j_{1},...,j_{n}}

and the multiple Wiener–Itô integral

I_{n}(f_{j_{1},k_{1},...,j_{n},k_{n}}|j_{1}, …, j_{n})
= ∫ f_{j_{1},k_{1},...,j_{n},k_{n}}(x_{1}, …, x_{n})Z_{G,j_{1}}( dx_{1}) … Z_{G,j_{n}}( dx_{n}).

The identity

: P (ξ_{j_{s},k_{s}}, 1 ≤ j_{s} ≤ d, 1 ≤ k_{s} ≤ m for all 1 ≤ s ≤ n):

= ∑_{1≤j_{s}≤d for all 1≤s≤n 1≤k_{s}≤m for all 1≤s≤n} a_{j_{1},k_{1},...j_{n},k_{n}}I_{n}(f_{j_{1},k_{1},...,j_{n},k_{n}}|j_{1}, …, j_{n})

holds.

Remark. Theorem 4.7 of [8] contains a version of this result for scalar-valued stationary Gaussian random fields.

Proof of the Corollary of Theorem 2.2. By Theorem 2.2 we have

a_{j_{1},k_{1},...j_{n},k_{n}}: ξ_{j_{1},k_{1}}ξ_{j_{2},k_{2}} ⋯ ξ_{j_{n},k_{n}}: = a_{j_{1},k_{1},...j_{n},k_{n}}I_{n}(f_{j_{1},k_{1},...,j_{n},k_{n}}|j_{1}, …, j_{n}) for all sequences of indices (j_{s}, k_{s}), 1 ≤ s ≤ n. By summing up these inequalities for all sequences of indices we get the proof of the corollary. □

With the help of the above corollary, we prove the following result.

Proposition 2.3. For all n ≥ 1 and functions f ∈ K_{n,j_{1},...,j_{n}} with some indices 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ n, I_{n}(f|j_{1}, …, j_{n}) ∈ H_{n} for the n-fold Wiener– Itô integral I_{n}(f|j_{1}, …, j_{n}). Besides, the set of all sums of n-fold Wiener–Itô integrals i.e., the set of all sums of the form

∑_{1≤j_{s}≤d for all 1≤s≤n} I_{n}(f_{j_{1},...,j_{n}}|j_{1}, …, j_{n}),

where f_{j_{1},...,j_{n}} ∈ K_{n,j_{1},...,j_{n}} constitute an everywhere dense linear subspace of H_{n}.

Proof of Proposition 2.3. We shall prove Proposition 2.3 by induction with respect to n. By Lemma 3.2 of [10] Proposition 2.3 holds for n = 1. Indeed, by this result every random variable of the form ξ = ∑_{j=1}^{d} ξ_{j} with some ξ_{j} ∈ H_{1,j} can be written as the sum of one-fold Wiener–Itô integrals, and the random variables of this form constitute an everywhere dense linear subspace of H_{1}.

If the statements of Proposition 2.3 hold for all m < n, then we can say for one part that I_{n}(f|j_{1}, …, j_{n}) ∈ H_{≤n}, because this relation holds if f is a simple function, i.e., if f ∈ K̂_{n,j_{1},...,j_{n}} with the space K̂_{n,j_{1},...,j_{n}} defined in Section 5 of [10], and since K̂_{n,j_{1},...,j_{n}} is dense in K_{n,j_{1},...,j_{n}}, and we defined the multiple Wiener–Itô integral by the extension of a bounded operator in the general case, the above property remains valid for general functions f ∈ K_{n,j_{1},...,j_{n}}. Moreover, we know that I_{n}(f|j_{1}, …, j_{n}) is orthogonal to all multiple Wiener–Itô integrals of the form I_{m}(h|j′_{1}, …, j′_{m}) with m < n because of relation (5.5) in [10]. Then I_{n}(f|j_{1}, …, j_{n}) is also orthogonal to any linear combination of such integrals. But these linear combinations constitute an everywhere dense set in H_{m} by our inductive hypothesis. Hence I_{n}(f|j_{1}, …, j_{n}) is orthogonal to the whole space H_{m} for all 0 ≤ m ≤ n − 1, and this implies that it is contained in the Hilbert subspace H_{n} (and not only in H_{≤n}). It follows from the corollary of Theorem 2.2 that the sums of multiple Wiener–Itô integrals considered in Proposition 2.3 are dense in H_{n}, and they constitute a linear subspace. Indeed, this corollary implies that a large class of Wick polynomials of order n can be expressed as a sum of such integrals, and the class of these Wick polynomials of order n is dense in H_{n}. Proposition 2.3 is proved. □

Remark. In Proposition 2.3 we expressed a dense subset of H_{n} as a sum of n-fold Wiener–Itô integrals, but we did not express all elements of H_{n} in such a form. But even this weaker result suffices for our purposes.

In the case of scalar-valued stationary random fields, we have a stronger result. In that case, we can express all elements of H_{n} as an n-fold Wiener– Itô integral, and actually, we can say somewhat more. There is a so-called Fock space representation of all elements h ∈ H, which represents the elements h ∈ H in the form of a sum of multiple Wiener–Itô integrals of different multiplicity. (See Theorem 4.2 of [8] together with the definition of Fock spaces on page 28 of [8].) Moreover, this result has some useful consequences about the properties of this representation.

We cannot prove a similar result in the vector-valued case. This difference appears because of the following reason. If a sequence of random variables h_{N} ∈ H_{n}, N = 0, 1, 2, …, has the property that h_{N} → h_{0} with some h_{0} ∈ H_{n} in the norm of H_{n} as N → ∞ in the scalar-valued case, then these random variables h_{N} can be expressed as n-fold Wiener–Itô integrals of such functions k_{N} ∈ K_{n} for which k_{N} → k_{0} in the norm of K_{n}. On the other hands, in the case of vector-valued models we do not have a similar result.

Next, we consider a vector-valued stationary Gaussian random field

X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν},

whose elements can be written in the form X_{j}(p) = ∫ e^{i(p,x)}Z_{G,j}( dx) by means of the random spectral measure Z_{G} = (Z_{G,1}, …, Z_{G,d}) of this random field for all p ∈ ℤ^{ν} and 1 ≤ j ≤ d. Let us consider a random variable Y ∈ H_{n} which can be represented as the n-fold Wiener–Itô integral of some function h ∈ K_{n,j_{1},...,j_{n}}, i.e.,

Y = ∫ h(x_{1}, …, x_{n})Z_{G,j_{1}}( dx_{1}) … Z_{G,j_{n}}( dx_{n}).  (2.6)

I shall express the shift transforms T_{u}Y, u ∈ ℤ^{ν}, of Y given in formula (2.6) by an expression which shows some similarity to the Fourier transform of a function.

To do this first I recall the definition of the shift transforms T_{u}, u ∈ ℤ^{ν}, in a stationary random field X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν}.

Given some element X_{j}(m), m ∈ ℤ^{ν}, 1 ≤ j ≤ d, of the random field, and u ∈ ℤ^{ν}, we define the shift transform of X_{j}(m) by T_{u} as T_{u}X_{j}(m) = X_{j}(u + m). More generally, given any measurable function h(X_{j}(m), m ∈ ℤ^{ν}, 1 ≤ j ≤ d), we define the shift transform of the random variable Y = h(X_{j}(m), m ∈ ℤ^{ν}, 1 ≤ j ≤ d), by the formula T_{u}Y = h(X_{j}(m+u), m ∈ ℤ^{ν}, 1 ≤ j ≤ d). This transformation was discussed in the scalar-valued case in [8]. It can be seen, (similarly to the argument in that work) that the definition of this transformation is meaningful, (i.e., the value of T_{u}Y does not depend on the choice of the function h for which Y = h(X_{j}(m), m ∈ ℤ^{ν}, 1 ≤ j ≤ d)), and we have defined in such a way unitary (linear) transformations T_{u}, u ∈ ℤ^{ν}, on H for which T_{u}T_{v} = T_{u+v}.

In Lemma 3.2 of [10] I have shown that each random variable U_{j} ∈ H_{1,j} can be written in the form U_{j} = ∫ h(x)Z_{G,j}( dx) with some function h(x) ∈ K_{1,j}. On the other hand, I claim that for all u ∈ ℤ^{ν}

T_{u}U_{j} = ∫ e^{i(u,x)}h(x)Z_{G,j}( dx) if U_{j} = ∫ h(x)Z_{G,j}( dx)  (2.7)

with some h ∈ K_{1,j}. Indeed, relation (2.7) clearly holds if h(x) = e^{i(p,x)} with some p ∈ ℤ^{ν}, since in this case U_{j} = X_{j}(p) and T_{u}U_{j} = X_{j}(p + u). But this implies that relation (2.7) holds for all finite trigonometrical polynomials of the form h(x) = ∑ c_{k}e^{i(p_{k},x)}, and for the closure of these functions with respect to the L_{2} norm determined by the measure G_{j,j}, i.e., for all h ∈ K_{1,j}.

In Proposition 2.4 I present a similar formula about the shift transforms of a random variable Y given by formula (2.6). This result is useful in the study of limit theorems related to non-linear functionals of a stationary Gaussian field.

Proposition 2.4 about the representation of shift transformations. Let a vector-valued stationary Gaussian random field

X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν},

be given with a vector-valued random spectral measure Z_{G} = (Z_{G,1}, …, Z_{G,d}) such that X_{j}(p) = ∫ e^{i(p,x)}Z_{G,j}( dx) for all p ∈ ℤ^{ν} and 1 ≤ j ≤ d. Let Y ∈ H_{n} be the random variable defined in formula (2.6) with the help of this vector-valued random spectral measure Z_{G} and some function h ∈ K_{n,j_{1},...,j_{n}}. Then

T_{u}Y = ∫ e^{i(u,x_{1}+···+x_{n})}h(x_{1}, …, x_{n})Z_{G,j_{1}}( dx_{1}) … Z_{G,j_{n}}( dx_{n})  (2.8)

for all u ∈ ℤ^{ν}.

Proof of Proposition 2.4. Formula (2.8) holds in the special case if n = 1, and h(x) ∈ K_{1,j}, since in this case Y = ∫ h(x)Z_{G,j}( dx), and

T_{u}Y = ∫ e^{i(u,x)}h(x)Z_{G,j}( dx)

by formula (2.7).

I claim that formula (2.8) also holds in the case when the random variable Y is given by formula (2.6) with a kernel function of the form h(x_{1}, …, x_{n}) = φ_{1}(x_{1}) ⋯ φ_{n}(x_{n}) defined with the help of some functions φ_{s}(x) ∈ K_{1,j_{s}}, 1 ≤ s ≤ n. Indeed, in this case Y =: U_{1} ⋯ U_{n}: with U_{s} = ∫ φ_{s}(x)Z_{G,j_{s}}( dx), 1 ≤ s ≤ n, because of Theorem 2.2. On the other hand, I claim that

T_{u}: U_{1} ⋯ U_{n}: =: (T_{u}U_{1}) ⋯ (T_{u}U_{n}): .

To see this let us observe that by Theorem 2B : U_{1} ⋯ U_{n}: is the orthogonal projection of the product U_{1} ⋯ U_{n} to the Hilbert subspace H_{n}(U_{1}, …, U_{n}). Similarly, : (T_{u}U_{1}) ⋯ (T_{u}U_{n}): is the orthogonal projection of the vector (T_{u}U_{1}) ⋯ (T_{u}U_{n}) to the Hilbert subspace H_{n}(T_{u}U_{1}, …, T_{u}U_{n}). Since the vectors (U_{1}, …, U_{n}) and (T_{u}U_{1}, …, T_{u}U_{n}) have the same distribution, and the Wick polynomial corresponding to their product can be calculated in the same way this implies that if : U_{1} ⋯ U_{n}: = g(U_{1}, …, U_{n}) with some function g, then

: (T_{u}U_{1}) ⋯ (T_{u}U_{n}): = g(T_{u}U_{1}, …, T_{u}U_{n})

with the same function g. (In the present case g(x_{1}, …, x_{n}) is a polynomial of order n.) On the other hand, T_{u}: U_{1} ⋯ U_{n}: = T_{u}g(U_{1}, …, U_{n}) = g(T_{u}U_{1}, …, T_{u}U_{n}) in this case. The above argument implies the desired identity.

Thus we can state that if Y is defined by formula (2.6) with a function

h(x_{1}, …, x_{n}) = φ_{1}(x_{1}) ⋯ φ_{n}(x_{n})

with the above properties, then

T_{u}Y = : (T_{u}U_{1}) ⋯ (T_{u}U_{n}):
= ∫ e^{i(u,x_{1}+···+x_{n})}h(x_{1}, …, x_{n})Z_{G,j_{1}}( dx_{1}) … Z_{G,j_{n}}( dx_{n})

because of Theorem 2.2 and the relation T_{u}U_{s} = ∫ e^{i(u,x)}φ_{s}(x)Z_{G,j_{s}}( dx) for all indices 1 ≤ s ≤ n.

From the result in the previous case follows that relation (2.8) also holds if Y is defined by (2.6) with a function h(x_{1}, …, x_{n}) of the form of a finite sum

h(x_{1}, …, x_{n}) = ∑_{k} φ_{1,k}(x_{1})φ_{2,k}(x_{2}) ⋯ φ_{n,k}(x_{n})

with φ_{s,k} ∈ K_{1,j_{s}}, 1 ≤ s ≤ n.

Since functions of the above form are dense in K_{n,j_{1},...,j_{n}}, T_{u} is a unitary operator, and both (linear) transformations

h(x_{1}, …, x_{n}) → e^{i(u,x_{1}+···+x_{n})}h(x_{1}, …, x_{n})

and h → I_{n}(h|j_{1}, …, j_{n}) from the space K_{n,j_{1},...,j_{n}} to the spaces K_{n,j_{1},...,j_{n}} and H_{n} are of bounded norms, it is not difficult to see that Proposition 2.4 holds in the general case. Proposition 2.4 is proved. □

3. On the proof of limit theorems for non-linear functionals of vector-valued stationary Gaussian random fields

First I recall the limit theorem problem we are interested in.

Let X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν}, be a d-dimensional vector-valued Gaussian stationary field, and let a function H(x_{1}, …, x_{d}), H: ℝ^{d} → ℝ^{1}, of d variables be given. Let us define with their help the random variables Y (p) = H(X_{1}(p), …, X_{d}(p)) for all p ∈ ℤ^{ν}, and introduce for all N = 1, 2, … the normalized random sum

S_{N} = A_{N}^{−1} ∑_{p∈B_{N}} Y (p)  (3.1)

with an appropriate norming constant A_{N} > 0, where

B_{N} = {p = (p_{1}, …, p_{ν}): 0 ≤ p_{k} < N for all 1 ≤ k ≤ ν}.  (3.2)

Let us also fix the vector-valued random spectral measure (Z_{G,1}, …, Z_{G,d}) on the torus [−π, π)^{ν} for which X_{j}(p) = ∫ e^{i(p,x)}Z_{G,j}( dx), 1 ≤ j ≤ d, p ∈ ℤ^{ν}. We are interested in the question what kind of limit theorems may hold for the normalized sums S_{N} defined in (3.1) as N → ∞ with an appropriate norming constants A_{N}. Here we are interested in the case when the correlation functions r_{j,j′}(p) = EX_{j}(0)X_{j′}(p), 1 ≤ j, j′ ≤ d, tend to zero slowly as |p| → ∞. This means strong dependence of the random variables in the stationary random fields. In such cases, we can get limit theorems with a non-Gaussian limit.

We have studied the above problem in [5] for scalar-valued stationary random fields, i.e., in the case d = 1, and we have proved some new kinds of limit theorems. Let me remark that at the same time M. Taqqu also proved similar results with the help of a different method, see [13]. I do not discuss Taqqu’s work, because here I am interested in the question of how to generalize the method in [5] to prove limit theorems also for non-linear functionals of vector-valued stationary Gaussian random fields.

In paper [10] and Section 2 of this work I discussed the notions and results we have to adopt some important methods of [8] when we are working with multivariate models. In this section, I explain how to generalize those methods of [8] which lead to non-central limit theorems when we are working with non-linear functionals of vector-valued stationary Gaussian random fields. I shall give the proof of the limit theorems in paper [11] with the help of the results in this paper.

In the first step of this discussion I rewrite the limit problem we are interested in in a different form. Let us observe that we have X_{j}(p) = T_{p}X_{j}(0) with the shift transform T_{p} for all p ∈ ℤ^{ν} and 1 ≤ j ≤ d, hence Y (p) = T_{p}Y (0), and we can rewrite the sum in (3.1) in the form

S_{N} = A_{N}^{−1} ∑_{p∈B_{N}} T_{p}Y (0).  (3.3)

As it will turn out the crucial point in the investigation of our limit theorems is the study of limit theorems in the special case when Y (0) is a Wick polynomial, and here we restrict our attention to this case.

Let us consider the case when Y (0) is a Wick polynomial of order k which has the form

Y (0) = : ∑_{\begin{matrix}(k_{1},...,k_{d}) \\ k_{1}+···+k_{d}=k \end{matrix}} a_{k_{1},...,k_{d}}X_{1}(0)^{k_{1}} ⋯ X_{d}(0)^{k_{d}}:  (3.4)

with some real coefficients a_{k_{1},...,k_{d}}. Then by the corollary of Theorem 2.2 and the identities X_{j}((0) = ∫ 𝕀_{1}(x)Z_{G,j}( dx), 1 ≤ j ≤ d, where 𝕀_{1}(·) denotes the indicator function of the torus [−π, π)^{ν}, the random variable Y (0) can be written in the form

Y (0) = ∑_{\begin{matrix}(k_{1},...,k_{d}) \\ k_{j}≥0, 1≤j≤d, \\ k_{1}+···+k_{d}=k \end{matrix}}a_{k_{1},...,k_{d}} ∫ 𝕀_{1}(x_{1}) … 𝕀_{1}(x_{k}) ∏_{j=1}^{d} ( ∏_{t=k_{1}+···+k_{j−1}+1}^{k_{1}+···+k_{j}}Z_{G,j}( dx_{t})) ,

where for j = 1 we define ∏_{t=k_{1}+···+k_{j−1}+1}^{k_{1}+···+k_{j}} Z_{G,j}( dx_{t}) = ∏_{t=1}^{k_{1}} Z_{G,1}( dx_{t}), and if

k_{j} = 0 for some 1 ≤ j ≤ d, then the product ∏_{t=k_{1}+···+k_{j−1}+1}^{k_{1}+···+k_{j}} Z_{G_{j}}( dx_{t}) is

omitted from this expression.

By Proposition 2.4 we can write

T_{p}Y (0) = ∑_{\begin{matrix}(k_{1},...,k_{d}) \\ k_{j}≥0, 1≤j≤d, \\ k_{1}+···+k_{d}=k \end{matrix}}a_{k_{1},...,k_{d}} ∫ e^{i(p,x_{1}+···+x_{k})} ∏_{j=1}^{d} ( ∏_{t=k_{1}+···+k_{j−1}+1}^{k_{1}+···+k_{j}} Z_{G.j}( dx_{t}))

(3.5)

for all indices p ∈ ℤ^{ν}.

We get by summing up formula (3.5) for all p ∈ B_{N} with our choice of Y (0) that

S_{N} = A_{N}^{−1} ∑_{\begin{matrix}(k_{1},...,k_{d}) \\ k_{j}≥0, 1≤j≤d, \\ k_{1}+···+k_{d}=k \end{matrix}} a_{k_{1},...,k_{d}} ∫  ∏_{l=1}^{ν} \frac{e^{iN(x_{1}^{(l)}+···+x_{k}^{(l)})} − 1}{e^{i(x_{1}^{(l)}+···+x_{k}^{(l)})} − 1}

∏_{j=1}^{d} ( ∏_{t=k_{1}+···+k_{j−1}+1}^{k_{1}+···+k_{j}} Z_{G,j}( dx_{t})) ,

where we write x = (x^{(1)}, …, x^{(ν)}) for all x ∈ [−π, π)^{ν}. (The set B_{N} was defined in (3.2).) I shall rewrite the above identity in a form more appropriate for us, First I apply the change of variables y_{l} = Nx_{l}, 1 ≤ l ≤ n. It yields that

S_{N} = ∑_{\begin{matrix}(k_{1},...,k_{d}) \\ k_{j}≥0, 1≤j≤d, \\ k_{1}+···+k_{d}=k \end{matrix}} ∫ h_{k_{1},...,k_{d}}^{N}(y_{1}, …, y_{k}) ∏_{j=1}^{d} ( ∏_{t=k_{1}+···+k_{j−1}+1}^{k_{1}+···+k_{j}} Z_{G^{(N)},j}( dy_{t})) ,

(3.6)

where

h_{k_{1},...,k_{d}}^{N}(y_{1}, …, y_{k}) = a_{k_{1},...,k_{d}} ∏_{l=1}^{ν} \frac{e^{i(y_{1}^{(l)}+···+y_{k}^{(l)})} − 1}{N(e^{i(y_{1}^{(l)}+···+y_{k}^{(l)})/N} − 1)}

is a function on [−Nπ, Nπ)^{ν}, and Z_{G^{(N)},j}(A) = N^{ν/k}A_{N}^{−1/n}Z_{G,j}(\frac{A}{N}) is defined for all measurable sets A ⊂ [−Nπ, Nπ)^{ν} and j = 1, …, d. Here we use the notation y_{s} = (y_{s}^{(1)}, …, y_{s}^{(ν)}), 1 ≤ s ≤ k. Let us observe that (Z_{G^{(N)},1}, …, Z_{G^{(N)},d}) is a vector-valued random spectral measure on the torus [−Nπ, Nπ)^{ν}, corresponding to the matrix valued spectral measure G^{(N)} = (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, on the torus [−Nπ, Nπ)^{ν}), defined by the formula

G_{j,j′}^{(N)}(A) = N^{2ν/k}A_{N}^{−2/n}G_{j,j′} (\frac{A}{N} ) , 1 ≤ j, j′ ≤ d,

for all measurable sets A ⊂ [−Nπ, Nπ)^{ν}, where G = (G_{j,j′}), 1 ≤ j, j′ ≤ d, is the matrix valued spectral measure of the original vector-valued stationary random field X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν}. On the other hand, h_{k_{1},...,k_{d}}^{N} ∈ K_{k,j_{1},...,j_{k}}(G_{j_{1},j_{1}}^{(N)}, …, G_{j_{k},j_{k}}^{(N)}) with j_{p} = s if k_{1} + ⋯ + k_{s−1} < p ≤ k_{1} + ⋯ + k_{s}, 1 ≤ s ≤ d. (For s = 1 we define k_{1} + ⋯ + k_{s−1} = 0.)

In formula (3.6) we have taken summation for the series (k_{1}, …, k_{d}) ∈ K, where K = {(k_{1}, …, k_{d}): k_{s} ≥ 0, 1 ≤ s ≤ d, k_{1} + ⋯ + k_{d} = k}, and in each term of the sum at its right-hand side there was a product of the form ∏_{s=1}^{k} Z_{G^{(N)},j_{s}}(du_{t}) with a sequence (j_{1}, …, j_{k}) ∈ J, where J = {(j_{1}, …, j_{k}): 1 ≤ j_{1} ≤ j_{2} ≤ ⋯ ≤ j_{k} ≤ d}, and it depended on the sequence (k_{1}, …, k_{d}) which indices j_{1}, …, j_{k}) appeared in the above product.

We want to rewrite expression (3.6) in a more pleasant form. For this goal, we make the following observation. There is a natural invertible map T: J → K, defined as T (j_{1}, …, j_{k}) = (k_{1}(j_{1}, …, j_{k}), … k_{d}(j_{1}, …, j_{k})) for all (j_{1}, …, j_{k}) ∈ J, where for all indices 1 ≤ s ≤ d k_{s}(j_{1}, …, j_{k}) equals the number of those elements j_{p} in the sequence (j_{1}, …, j_{k}) for which j_{p} = s. Its inverse is defined by the formula

T^{−1}(k_{1}, …, k_{d}) = (j_{1}(k_{1}, …, k_{d}), …, j_{k}(k_{1}, …, k_{d}))

for all (k_{1}, …, k_{d}) ∈ K, where j_{s}(k_{1}, …, k_{d}) = min{u: k_{1} + ⋯ + k_{u} ≥ s}.

With the help of the above defined functions k_{s} = k_{s}(j_{1}, …, j_{k}), 1 ≤ s ≤ d, we can rewrite the identity in (3.6) in a form more appropriate for us. In this new formula we take summation for (j_{1}, …, j_{k}) ∈ J instead of (k_{1}, …, k_{d}) ∈ K. We get that

S_{N} = ∑_{\begin{matrix}(j_{1},...,j_{k}), \\ 1≤j_{1}≤···≤j_{k}≤d \end{matrix}} ∫ h_{j_{1},...j_{k}}^{N}(y_{1}, …, y_{k})Z_{G^{(N)},j_{1}}( dy_{1}) … Z_{G^{(N)},j_{k}}( dy_{k})

(3.7)

with

h_{j_{1},...,j_{k}}^{N}(y_{1}, …, y_{k}) = a_{k_{1}(j_{1},...,j_{k}),...,k_{d}(j_{1},...,j_{k})} ∏_{l=1}^{ν} \frac{e^{i(y_{1}^{(l)}+···+y_{k}^{(l)})} − 1}{N(e^{i(y_{1}^{(l)}+···+y_{k}^{(l)})/N} − 1)}.

Let us observe that

lim_{N→∞} h_{j_{1},...,j_{k}}^{N}(y_{1}, …, y_{k}) = h_{j_{1},...,_{k}}^{0}(y_{1}, …, y_{k})

with the function

h_{j_{1},...,j_{k}}^{0}(y_{1}, …, y_{k}) = a_{k_{1}(j_{1},...,j_{k}),...,k_{d}(j_{1},...,j_{k})} ∏_{l=1}^{ν} \frac{e^{i(y_{1}^{(l)}+···+y_{k}^{(l)})} − 1}{i(y_{1}^{(l)} + ⋯ + y_{k}^{(l)})}

defined on ℝ^{kν}, and this convergence is uniform in all bounded subsets of ℝ^{kν}.

It is natural to expect that if the matrix valued spectral measures G^{(N)} = (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, converge to a matrix valued spectral measure G^{(0)} =

(G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, defined on ℝ^{ν} in an appropriate way, then a limiting procedure in formula (3.7) supplies the limit theorem S_{N} → S_{0} in distribution with

S_{0} = ∑_{\begin{matrix}(j_{1},...,j_{k}) \\ 1≤j_{1}≤···≤j_{k}≤d \end{matrix}} ∫ h_{j_{1},...j_{k}}^{0}(y_{1}, …, y_{k})Z_{G^{(0)},j_{1}}( dy_{1}) … Z_{G^{(0)},j_{k}}( dy_{k})

as N → ∞, where (Z_{G^{(0)},1}, …, Z_{G^{(0)},d}) is a vector-valued random spectral

measure on ℝ^{ν} corresponding to the matrix valued spectral measure (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d. On the other hand, the convergence of the spectral measures G^{(N)} to the spectral measure G^{(0)} is satisfied in many interesting models, for instance in the models investigated in paper [11].

Next, I explain how to work out a precise method to prove limit theorems on the basis of the above heuristic argument. In particular, we are interested in the question of when the above sketched heuristic argument can be carried out. In the scalar-valued case, this problem was solved in Lemma 8.3 of [8]. Here I prove the vector-valued variant of this result.

In the formulation of Lemma 8.3 of [8], we had to introduce a version of the notion of weak convergence of finite measures to a larger class of measures, to the class of so-called locally finite measures. They are measures, whose restrictions to any compact set are finite. Here I introduce a slight generalization of the notion called vague convergence in [8] to the case when we are working with complex measures of locally finite total variation. In [8] we have worked with (positive) measures. (The definition of complex measures on ℝ^{ν} with locally finite total variation was explained in Section 4 of the paper [10].)

Definition of vague convergence of complex measures on ℝ^{ν} with locally finite total variation. Let G_{N}, N = 1, 2, …, be a sequence of complex measures on ℝ^{ν} with locally finite total variation. We say that the sequence G_{N} vaguely converges to a complex measure G_{0} on ℝ^{ν} with locally finite total variation (in notation G_{N} →^{v} G_{0}) if

lim_{N→∞} ∫ f(x) G_{N}( dx) = ∫ f(x) G_{0}( dx)  (3.8)

for all continuous functions f on ℝ^{ν} with a bounded support.

I shall take a sequence of sums of k-fold Wiener–Itô integrals, and then I formulate Proposition 3.1 which states that under some appropriate conditions these sums of random integrals have a limit that can be expressed in an explicit form. This result together with the representation of non-linear functionals of vector-valued stationary Gaussian random fields by means of multiple Wiener–Itô integrals enable us to prove limit theorems with a non-Gaussian limit for non-linear functionals of vector-valued stationary Gaussian random fields.

For all N = 1, 2, … take a sequence of matrix valued non-atomic spectral measures (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, on the torus [−A_{N}π, A_{N}π)^{ν} with parameter A_{N} such that A_{N} → ∞ as N → ∞. Let us also take some functions

h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) ∈ K_{k,j_{1},...,j_{k}} = K_{k,j_{1},...,j_{k}}(G_{j_{1},j_{1}}^{(N)}, …, G_{j_{k},j_{k}}^{(N)})

on the torus [−A_{N}π, A_{N}π)^{ν} for all (j_{1}, …, j_{k}) with 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, and N = 1, 2, . . .. For all N = 1, 2, … fix a vector-valued random spectral measure

(Z_{G^{(N)},1}, …, Z_{G^{(N)},d}^{(N)})

on the torus [−A_{N}π, A_{N}π)^{ν} corresponding to the matrix valued spectral measure (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d. Let us define with the help of these quantities the sums of n-fold Wiener–Itô integrals

Z_{N} = ∑_{\begin{matrix}(j_{1},...,j_{k}) \\ 1≤j_{s}≤d for all 1≤s≤k \end{matrix}}∫ h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k})Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k}),

(3.9)

N = 1, 2, . . .. In the next result I show that under appropriate conditions these random variables Z_{N} converge in distribution to a random variable Z_{0} expressed in the form of a sum of multiple Wiener–Itô integrals.

Proposition 3.1. For all N = 1, 2, … let us consider the sums of k-fold Wiener–Itô integrals Z_{N} defined in formula (3.9) with the help of certain vector-valued random spectral measures (Z_{G^{(N)},1}, …, Z_{G^{(N)},d}) corresponding

to some non-atomic matrix valued spectral measures (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, defined on tori [−A_{N}, A_{N})^{ν} such that A_{N} → ∞ as N → ∞, and functions

h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) ∈ K_{k,j_{1},...,j_{k}}(G_{j_{1},j_{1}}^{(N)}, …, G_{j_{k},j_{k}}^{(N)}).

Let the coordinates G_{j,j′}^{(N)}, 1 ≤ j, j′ ≤ d, of the matrix valued spectral mea-

sures (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, converge vaguely to the coordinates G_{j,j′}^{(0)} of a

non-atomic matrix valued spectral measure (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, on ℝ^{ν} for all 1 ≤ j, j′ ≤ d as N → ∞, and let (Z_{G^{(0)},1}, …, Z_{G^{(0)},d}) be a vector-valued random spectral measure on ℝ^{ν} corresponding to the matrix valued spectral measure (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d. Let us also have some functions h_{j_{1},...,j_{k}}^{0} for all (j_{1}, …, j_{k}) with 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, such that these functions and matrix valued spectral measures satisfy the following conditions.

(a): The functions h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k}) are continuous on ℝ^{kν} for all 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, and for all T > 0 and indices 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, the functions h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) converge uniformly to the function h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k}) on the cube [−T, T]^{kν} as N → ∞.

(b): For all ε > 0 there is some T_{0} = T_{0}(ε) > 0 such that

∫_{ℝ^{kν}\[−T,T]^{kν}} |h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k})|^{2}G_{j_{1},j_{1}}^{(N)}( dx_{1}) … G_{j_{k},j_{k}}^{(N)}(dx_{k}) < ε^{2} (3.10)
for all 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, and N = 1, 2 … if T > T_{0}.

Then inequality (3.10) holds also for N = 0,

h_{j_{1},...,j_{k}}^{0} ∈ K_{k,j_{1},...,j_{k}} = K_{k,j_{1},...,j_{k}}(G_{j_{1},j_{1}}^{(0)}, … G_{j_{k},j_{k}}^{(0)}),  (3.11)

the sum of random integrals

Z_{0} = ∑_{\begin{matrix}(j_{1},...,j_{k}) \\ 1≤j_{s}≤d, for all 1≤s≤k \end{matrix}}∫ h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})Z_{G^{(0)},j_{1}}( dx_{1}) … Z_{G^{(0)},j_{k}}( dx_{k})

(3.12)

exists, and the random variables Z_{N} defined in (3.9) satisfy the relation Z_{N} →^{D} Z_{0} as N → ∞, where →^{D} denotes convergence in distribution.

Remark 1. A complex measure G_{j,j′}^{(N)} with finite total variation defined on the torus [−A_{N}π, A_{N}π)^{ν} can be identified in a natural way with a complex measure on ℝ^{ν} which is concentrated on its subset [−A_{N}π, A_{N}π)^{ν}. We take this identification of G_{j,j′}^{(N)} with a complex measure on ℝ^{ν} when we give

meaning to formula (3.8) with G_{N} = G_{j,j′}^{(N)} and G_{0} = G_{j,j′}^{(0)} in the definition

of the vague convergence of the complex measures G_{j,j′}^{(N)} to G_{j,j′}^{(0)} as N → ∞.

Remark 2. In Proposition 3.1 we imposed two conditions for the convergence of the sums of multiple Wiener–Itô integrals Z_{N} defined in (3.9) to the sum of multiple Wiener–Itô integrals Z_{0} defined in (3.12). First we demanded that the spectral measures and kernel functions appearing in the definition of the expressions Z_{N} should converge to the corresponding spectral measure and kernel functions appearing in the definition of the expression Z_{0} in an appropriate way. We still imposed an additional condition in part (b) of Proposition 3.1. This is a compactness type condition which implies that the essential part of the random integrals in the definition of the random variables Z_{N} are essentially concentrated in a compact set not depending on the parameter N. I shall show in the Appendix with the help of an example that condition (b) cannot be omitted from Proposition 3.1. If we drop it Proposition 3.1 may not hold any longer. I shall also make some additional remarks about Proposition 3.1.

Proposition 3.1 is a multivariate version of Lemma 8.3 in [8]. I gave a simpler proof of the scalar-valued result in Lemma 6.6 of [9], and I shall adopt this proof. In the proof of Proposition 3.1 we have to overcome some additional difficulties, which arose because we are working with vectorvalued random fields. First I explain the method of the proof.

In the first step of the proof, we show that relation (3.11) holds, and the random integrals appearing in the definition of Z_{0} really exist. The main step in the proof of this result is to show that the measures μ_{j_{1},...,j_{k}}^{(N)} defined at

the start of the proof weakly converge to a measure μ_{j_{1},...,j_{k}}^{(0)}. First, we prove the vague convergence of these measures with the help of the conditions at the beginning of Proposition 3.1, and then we show with the help of condition (b) of this result that weak convergence holds, too.

In the next step, we reduce the proof of Proposition 3.1 to that of relation (3.13). This reduction contains the statement of Proposition 3.1 in the special case when the kernel functions

h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) and h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})

in the random integrals in formulas (3.9) and (3.12) are replaced by

h_{j_{1},...,j_{n}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k}),

where χ_{T} (x_{1}, …, x_{k}) is the indicator function of the cube [−T, T]^{kν}.

The proof of this reduction is based on the application of inequality (5.6) in [10] which helps to show that the expressions Z_{N} in (3.9) and (3.12) are changing very little if we restrict the domain of integration in the Wiener– Itô integrals appearing in them to a cube [−T, T]^{kν} with a sufficiently large number T > 0 or if we replace the kernel functions h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) in these random integrals by their limit h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k}). We also exploit that the weak convergence of distribution functions to a limit can be reformulated by means of their characteristic functions in an equivalent way.

Then we formulate a new, simpler version of the limit relation (3.13) in formula (3.18) which implies relation (3.13), hence Proposition 3.1, too. In this version we replace the kernel functions h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k}) in (3.13) by such simple functions which provide a good approximation of them. (See Section 5 of [10] for the definition of simple functions and of their properties needed in our proof.) We want to find such simple functions which provide a good approximation of the random integrals appearing at the right-hand side of (3.13) if we replace each kernel function in these integrals by the appropriate element of them. First we want to guarantee only the weaker statement that we get a good approximation by this replacement in the special case when the expression at the right-hand side of (3.13) is considered. On the other hand, we prescribe an additional property of these simple functions. We demand that they should be adapted to such a regular system whose elements have boundaries with zero μ_{0} measure with respect to an appropriate measure μ_{0}. We impose this extra condition because it will imply that the replacement of the kernel functions in the random integrals in (3.13) with the appropriate simple functions provides a good approximation not only for the expression on the right-hand side of this formula, where we integrate with respect to the random spectral measures Z_{G^{(0)},j}, but also for the expressions on the left-hand side of this formula for large indices N, where we integrate with respect to the random spectral measures Z_{G^{(N)},j}.

The reduction of the proof of (3.13) to the proof of (3.18) is done similarly to the reduction of the proof of Proposition 3.1 to the proof of (3.13) with the help of formulas (3.16) and (3.17). These relations are consequences of the properties of the simple functions we chose to approximate the kernel functions in the random integrals in (3.13). Then we have to prove relation (3.18). At this point, some new difficulties arise if we are working with vector (and not scalar) valued random fields.

An essential point of the proof of (3.18) is to show that under the conditions of Proposition 3.1 G_{j,j′}^{(N)}(Δ) → G_{j,j′}^{(0)}(Δ) as N → ∞ for all 1 ≤ j, j′ ≤ d if the boundary of the set Δ behaves nicely. The proof is fairly simple for j = j′. In this case, we can apply some results about weak convergence of measures. But the proof is more difficult for pairs (j, j′) with j ≠ j′. This difficulty arises, because if j ≠ j′, then G_{j,j′}^{(N)}(·) with a fixed parameter N can be not a (real-valued, positive) measure. To get a proof in this case we apply a special argument, where we exploit that G_{j,j′}^{(N)} is one of the coordinates of a positive semidefinite matrix valued measure G^{(N)}.

If the limit behavior of the complex measures G_{j,j′}^{(N)} with locally finite total variation is already known, then relation (3.18) can be proved in a standard way. For all N = 0, 1, 2, … we consider the random vector Z_{N}(D) we obtain if we restrict the random spectral measure Z_{G^{N}} to the elements of the regular system D we are working with. It can be proved that the distributions of the random vectors Z_{N}(D) weakly converge to the distribution of Z_{0}(D) as N → ∞. As the random variables Y_{N} appearing in formula (3.18) are polynomials of the arguments in the random vectors Z_{N}(D), the limit relation in (3.18) simply follows from the above weak convergence.

Proof of Proposition 3.1. First I show that relation (3.10) holds also for N = 0. To see this let us first show that the measures μ_{j_{1},...,j_{k}}^{(N)}, N = 1, 2, …, defined as

μ_{j_{1},...,j_{k}}^{(N)}(A) = ∫_{A} |h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k})|^{2}G_{j_{1},j_{1}}^{(N)}( dx_{1}) … G_{j_{k},j_{k}}^{(N)}(dx_{k}), A ⊂ ℝ^{kν},

converge vaguely to the locally finite measure μ_{j_{1},...,j_{k}}^{(0)} defined as

μ_{j_{1},...,j_{k}}^{(0)}(A) = ∫_{A} |h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})|^{2}G_{j_{1},j_{1}}^{(0)}( dx_{1}) … G_{j_{k},j_{k}}^{(0)}(dx_{k}), A ⊂ ℝ^{kν},

if N → ∞.

Indeed, it follows from the vague convergence of the measures G_{j,j}^{(N)} to G_{j,j}^{(0)}

as N → ∞ and the continuity of the function h_{j_{1},...,j_{n}}^{(0)} that this relation holds if we replace the kernel function |h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k})|^{2} by the kernel function |h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})|^{2} in the definition of the measures μ_{j_{1},...,j_{k}}^{(N)}. Then condition (a) of Proposition 3.1 implies that this relation also holds with the original definition of the measures μ_{j_{1},...,j_{k}}^{(N)}.

Next I state that the measure μ_{j_{1},...,j_{k}}^{(0)} is finite, and the measures μ_{j_{1},...,j_{k}}^{(N)} converge to it not only vaguely but also weakly. Indeed, condition (b) implies that the sequence of measures μ_{j_{1},...,j_{k}}^{(N)} is compact with respect to the topology defining the weak convergence of finite measures, hence any subsequence of it has a convergent sub-subsequence. But the limit of such a sub-subsequence can be only its limit with respect to the vague convergence, i.e., it is μ_{j_{1},...,j_{k}}^{(0)}. This implies that μ_{j_{1},...,j_{k}}^{(0)} is a finite measure, and the se-

quence of measures μ_{j_{1},...,j_{k}}^{(N)} converges also weakly to it.

Finally the properties of the functions h_{j_{1},...,j_{k}}^{N}, and their convergence to h_{j_{1},...,j_{k}}^{0} formulated in condition (a) imply that also the symmetry property h_{j_{1},,...,j_{k}}^{0}(−x_{1}, …, −x_{k}) = \overline{h_{j_{1},,...,j_{k}}^{0}(x_{1}, …, x_{k})} holds, hence relation (3.11) is valid, and the random integral Z_{0} defined in (3.12) is meaningful.

Next I reduce the proof of the relation Z_{N} →^{D} Z_{0} to the proof of the following statement:

Under the conditions of Proposition 3.1

∑_{\begin{matrix}(j_{1},...,j_{k}) \\ 1≤j_{s}≤d, for all 1≤s≤k \end{matrix}}∫ h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k})  (3.13)

Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k})

→^{D} ∑_{\begin{matrix}(j_{1},...,j_{k}) \\ 1≤j_{s}≤d, for all 1≤s≤k \end{matrix}} ∫ h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k})

Z_{G^{(0)},j_{1}}( dx_{1}) … Z_{G^{(0)},j_{k}}( dx_{k}),

as N → ∞, where χ_{T} (x_{1}, …, x_{k}) is the indicator function of the cube [−T, T]^{kν}. We make a small, not so important technical restriction in the choice of the number T in (3.13). Let me recall that for all all vector-valued spectral measures G_{N} there is a finite measure μ_{N} on ℝ^{ν} such that all coordinates G_{j,j′}^{(N)} of G_{N}, (more precisely their restrictions to any compact sets) are such measures which are absolutely continuous measures with respect to μ_{N}. (See Section 4 of [10].) We fix such a measure μ_{N} for all N = 0, 1, 2, …, and we shall call them dominating measures. We shall work with such measures μ_{N} in the proof of Proposition 3.1. We state formula (3.13) for all such T > 0 for which the boundary of the cube [−T, T]^{kν} has zero measure with respect to the measure \underbrace{μ_{0} × ⋯ × μ_{0}}_{k times}.

To prove this reduction let us observe that by formulas (5.6) in [10] and (3.10)

E[∫ [1 − χ_{T} (x_{1}, …, x_{k})]h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k})

Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k})]^{2}

≤ k! ∫_{ℝ^{kν}\[−T,T]^{kν}} |h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{n})|^{2}G_{j_{1},j_{1}}^{(N)}( dx_{1}) … G_{j_{k},j_{k}}^{(N)}( dx_{k}) < k!ε^{2}

for all sequences (j_{1}, …, j_{k}), 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, and N = 0, 1, 2, … if T > T_{0}(ε). Hence

E[ ∑_{\begin{matrix}(j_{1},...,j_{k}) \\ 1≤j_{s}≤d for all 1≤s≤k \end{matrix}} ∫ [1 − χ_{T} (x_{1}, …, x_{k})]h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k})

Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k})]^{2} ≤ d^{k}k!ε^{2}  (3.14)

for all N = 0, 1, … if T > T_{0}(ε).

Since G_{j,j}^{(N)} →^{v} G_{j,j}^{(0)} for all 1 ≤ j ≤ d as N → ∞, hence for all T > 0 there

is some number C(T) such that G_{j,j}^{(N)}([−T, T]) ≤ C(T) for all N = 1, 2, … and 1 ≤ j ≤ d. Because of this estimate and the uniform convergence h_{j_{1},...,j_{k}}^{N} → h_{j_{1},...,j_{k}}^{0} on any cube [−T, T]^{kν} we have

E[∫ [h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) − h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})]χ_{T} (x_{1}, …, x_{k})

Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k})]^{2}

≤ k! ∫_{[−T,T]^{kν}} |h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) − h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})|^{2}
G_{j_{1},j_{1}}^{(N)}( dx_{1}) … G_{j_{k},j_{k}}^{(N)}( dx_{k}) < ε^{2}

for all T > 0 and (j_{1}, …, j_{k}), 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, if N > N_{1} with some N_{1} = N_{1}(T, ε). Hence

E[ ∑_{1≤j_{1},...,j_{k}≤d} ∫ [h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) − h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})] (3.15)

χ_{T} (x_{1}, …, x_{k})Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k})]^{2} ≤ d^{k}ε^{2}

for all T > 0 if N > N_{1} with some N_{1} = N_{1}(T, ε).

Let us define the quantities

U_{N} = U_{N}(T) = ∑_{1≤j_{1},...,j_{k}≤d} ∫  h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k})
Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k}),

and

V_{N} = V_{N}(T) = ∑_{1≤j_{1},...,j_{k}≤d} ∫  h_{j_{1},...,j_{n}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k})
Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k}),

N = 1, 2, . . .. We introduce the definition of V_{N} = V_{N}(T) also for N = 0, where we replace the spectral measures Z_{G^{(N)},j}, N ≥ 1, 1 ≤ j ≤ d, by Z_{G^{(0)},j}, 1 ≤ j ≤ d, in the definition of V_{N}. We can reduce the proof of the

relation Z_{N} →^{D} Z_{0} to formula (3.13) in the following way. By formula (3.14) we can state that

|E(e^{itZ_{N}} − e^{itU_{N}})| ≤ E|(1 − e^{it(Z_{N} −U_{N})})| ≤ E|(t(Z_{N} − U_{N})|
≤ |t|(E(Z_{N} − U_{N})^{2})^{1/2} ≤ |t|(d^{k}k!)^{1/2}ε.

for all t ∈ ℝ^{1} with the random variable Z_{N} defined in (3.9) if T > T_{0} and N > N_{0}(ε). Similarly, |E(e^{itU_{N}} − e^{itV_{N}})| ≤ |t|(E(U_{N} − V_{N})^{2})^{1/2} ≤ |t|d^{k/2}ε for all t ∈ ℝ^{1} and N > N_{0} by inequality (3.15). Besides, (3.14) with N = 0 implies that

E|e^{itV_{0}} − Ee^{itZ_{0}}| ≤ |t|(E(Z_{0} − V_{o})^{2})^{1/2} ≤ |t|d^{n/2}ε

for all t ∈ ℝ^{1} if T > T_{0}(ε), where Z_{0} is defined in (3.12) and V_{0} after the definition of V_{N} for N ≥ 1. Finally, Ee^{itV_{N}} → Ee^{itV_{0}} for all t ∈ ℝ^{1} if relation (3.13) holds. These relations together imply that |Ee^{itZ_{N}} − Ee^{itZ_{0}}| ≤ C(t)ε if N > N_{0}(t, ε) with some numbers C(t) and N_{0}(t, ε). Since this inequality holds for all ε > 0, it implies that Z_{N} →^{D} Z_{0}. (In formula (3.13) we imposed a condition on the parameter T > 0. We demanded that the boundary of [−T, T]^{kν} must have measure zero with respect to the product measure of μ_{0}. It causes no problem that we can apply the above argument only for parameters T with this property.)

We shall prove (3.13) with the help of some statements formulated below. To formulate them let us first fix a number T > 0 such that the boundary of the cube [−T, T]^{kν} has zero measure with respect to the measure \underbrace{μ_{0} × ⋯ × μ_{0}}_{n times}. Observe that

h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k}) ∈ K_{k,j_{1},...,j_{k}}(G_{j_{1},j_{1}}^{(N)}, …, G_{j_{k},j_{k}}^{(N)})

for all T > 0 and N = 0, 1, 2, . . .. I claim that for all ε > 0 a regular system D = D(ε) = {Δ_{k}, k = ±1, …, ±M} can be constructed for which all of its elements have zero measure with respect the a dominating measure μ_{0}, i.e. μ_{0}(∂Δ_{k}) = 0 for all 1 ≤ |k| ≤ M, Δ_{k} ⊂ [−T, T]^{ν} for all 1 ≤ |k| ≤ M, and such that there exist some simple functions

f_{j_{1},...,j_{k}}^{ε} ∈ K̂_{n,j_{1},...,j_{k}}(G_{j_{1},j_{1}}^{(0)}, …, G_{j_{n},j_{k}}^{(0)})

indexed by the parameters (j_{1}, …, j_{k}), 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, which are adapted to this regular system, and satisfy the inequalities written down in the following two formulas (3.16 and (3.17).

∫ |h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k}) − f_{j_{1},...,j_{k}}^{ε}(x_{1}, …, x_{k})|^{2}
G_{j_{1},j_{1}}^{(0)}( dx_{1}) … G_{j_{k},j_{k}}^{(0)}( dx_{k}) < ε^{2}  (3.16)

for all 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, and also

∫ |h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k}) − f_{j_{1},...,j_{k}}^{ε}(x_{1}, …, x_{k})|^{2}
G_{j_{1},j_{1}}^{(N)}( dx_{1}) … G_{j_{k},j_{k}}^{(N)}( dx_{k}) < ε^{2}  (3.17)

for all 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, and N ≥ N_{0} with some N_{0} = N_{0}(ε, T). I also claim that with such a choice of simple functions

Y_{N} →^{D} Y_{0}  (3.18)

as N → ∞, where

Y_{N} = Y_{N}(ε, T)
=  ∑_{\begin{matrix}(j_{1},...,j_{k}) \\ 1≤j_{s}≤d for all 1≤s≤k \end{matrix}} ∫ f_{j_{1},...,j_{k}}^{ε}(x_{1}, …, x_{k})Z_{G^{(N)},j_{1}}( dx_{1}) … Z_{G^{(N)},j_{k}}( dx_{k})

for N = 0, 1, 2, . . ..

Let us show that for all ε > 0 there exists a regular system D together with some simple functions f_{j_{1},...,j_{k}}^{ε} adapted to it which satisfy the desired properties.

Indeed, by Lemma (5.2) of [10] for all ε > 0 and parameters (j_{1}, …, j_{k}), 1 ≤ j_{s} ≤ d, 1 ≤ s ≤ k, there exists such a simple function f_{j_{1},...,j_{k}}^{ε} adapted to a regular system D_{j_{1},...,j_{k}} in such a way that this function f_{j_{1},...,j_{k}}^{ε} satisfies (3.16), and the elements of D_{j_{1},...,j_{k}} have boundaries with zero μ_{0} measure. Let us make such a construction for all parameters (j_{1}, …, j_{k}). It can be seen that there is a refinement D of the regular systems D_{j_{1},...,j_{k}} such that all simple functions f_{j_{1},...,j_{k}}^{ε} are adapted to it, and its elements have boundaries with zero μ_{0} probability. (I omit the details of this construction.) This regular system together with the functions f_{j_{1},...,j_{k}}^{ε} adapted to it satisy the desired requirements, because, as we shall see, their properties imply that these functions satisfy not only (3.16), but also (3.17).

Relation (3.13) can be proved with the help of relations (3.16), (3.17) and (3.18) similarly to the reduction of the relation Z_{N} →^{D} Z_{0} to formula (3.13). Indeed, one gets from inequalities (3.16), (5.6) in [10] and the definition of the quantities V_{N} and Y_{0}, by applying an argument similar to the proof of relation (3.14) that

E(V_{0} − Y_{0})^{2} ≤ k!k^{d}ε^{2},

and also

E(V_{N} − Y_{N})^{2} ≤ k^{d}k!ε^{2}

if N > N_{0}(ε, T) by (3.17) and (5.6) in [10].

Then we can show with the help of these relations similarly to the reduction of the relation Z_{N} →^{D} Z_{0} to formula (3.13) that |Ee^{itV_{N}} − Ee^{itY_{N}} | ≤ ε, |Ee^{itY_{N}} − Ee^{itY_{0}}| ≤ ε, and |Ee^{itY_{0}} − Ee^{itV_{0}}| ≤ ε if N > N_{0}(ε, t, T) with some threshold index N_{0}(ε, t, T). Here in the first and third inequality we apply the last two inequalities which were consequences of (3.16) and (3.17), while the second inequality follows from (3.18). Since these relations hold for all ε > 0 they imply that Ee^{itV_{N}} → Ee^{itV_{0}} for all t ∈ ℝ^{1} as N → ∞, i.e. V_{N} →^{D} V_{0} as N → ∞, and this is formula (3.13) written with a different notation.

It remains to prove (3.16), (3.17) and (3.18). We made such a construction of a regular system and simple functions adapted to it with the help of Lemma 5.2 in [10] which satisfy (3.16). Then formula (3.17) follows from some classical results about vague (and weak) convergence of measures. Since we are working in the proof of (3.17) in a cube [−T, T]^{kν} it is enough to know the results about weak convergence to carry out our arguments.

Let us first observe that since the restrictions of the measures G_{j,j}^{(N)} to

[−T, T]^{ν} tend weakly to the restriction of the measure G_{j,j}^{(0)} to the cube [−T, T]^{ν} as N → ∞, we can also say that the restrictions of the product measures G_{j_{1},j_{1}}^{(N)} × ⋯ × G_{j_{k},j_{k}}^{(N)} to the cube [−T, T]^{kν} converge weakly to the

restriction of the product measure G_{j_{1},j_{1}}^{(0)} ×⋯×G_{j_{k},j_{k}}^{(0)} on the cube [−T, T]^{kν}, as N → ∞. On the other hand, the function

H_{j_{1},...j_{k}}^{0}(x_{1}, …, x_{k})
= |h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k})χ_{T} (x_{1}, …, x_{k}) − f_{j_{1},...,j_{k}}^{ε}(x_{1}, …, x_{k})|^{2}

is almost everywhere continuous with respect to the measure G_{j_{1},j_{1}}^{(0)} × ⋯ ×

G_{j_{k},j_{k}}^{(0)}. By the general theory about convergence of measures these properties imply that

∫ H_{j_{1},...j_{k}}^{0}(x_{1}, …, x_{k})G_{j_{1},j_{1}}^{(N)}( dx_{1}) … G_{j_{k},j_{k}}^{(N)}( dx_{k})
→ ∫ H_{j_{1},...j_{k}}^{0}(x_{1}, …, x_{k})G_{j_{1},j_{1}}^{(0)}( dx_{1}) … G_{j_{k},j_{k}}^{(0)}( dx_{k})

as N → ∞. (Such a convergence is proved for probability measures for example in [2].) A careful analysis shows that this result remains valid for sequences of finite, but not necessarily probability measures. Let me

32 PÉTER MAJOR

remark that here we are working with (real, non-negative) measures. The last relation together with (3.16) imply (3.17).

To prove relation (3.18) first we show that G_{j,j′}^{(N)}(Δ_{k}) → G_{j,j′}^{(0)}(Δ_{k}) as N → ∞ for all 1 ≤ j, j′ ≤ d and Δ_{k} ∈ D with the regular system D we are working with. (Let me recall that the boundary of all sets Δ_{k} ∈ D has zero μ_{0} measure and hence also zero G_{j,j′}^{(0)} measure.)

If j = j′ then this relation follows immediately from the facts that G_{j,j}^{(N)} →^{v} G_{j,j}^{(0)}, G_{j,j}^{(0)}(∂Δ_{k}) = 0 for all 1 ≤ |k| ≤ M, and G_{j,j}^{(N)} is a locally finite measure for all N = 0, 1, 2, . . .. If j ≠ j′, then we have to apply a more refined argument, since in this case we only know that G_{j,j′}^{(N)} is a complex measure with locally finite total variation. In this case we will exploit that the matrix valued measures (G_{j.j′}^{(N)}), 1 ≤ j, j′ ≤ d, are positive semidefinite. This implies that the Radon–Nikodym derivatives g_{j,j′}^{(N)} of the complex measures G_{j,j′}^{(N)} with respect to the dominating measure μ_{N} have the following property. For all N = 0, 1, 2, … and 1 ≤ j, j′ ≤ d such that j ≠ j′ the 2 × 2 matrices

g^{(N)}(x|j, j′) = ( \begin{matrix}g_{j,j}^{(N)}(x), g_{j,j′}^{(N)}(x) \\ g_{j′,j}^{(N)}(x), ng_{j′,j′}^{(N)}(x) \end{matrix} )

are positive semidefinite for μ_{N} almost all x ∈ ℝ^{ν}. Let us define for all non-negative functions v(x), x ∈ ℝ^{ν} the vector S(x|v) = (\sqrt{v(x)}, \sqrt{v(x)}). By exploiting that the matrices g^{(N)}(x|j, j′) are positive semidefinite we get that

∫ v(x)[G_{j,j}^{(N)}( dx) + G_{j,j′}^{(N)}( dx) + G_{j′,j}^{(N)}( dx) + G_{j′,j′}^{(N)}( dx)]
= ∫ S(x|v)g^{(N)}(x|j, j′)S(x|v)^{∗}μ_{N}( dx) ≥ 0

for all functions v such that v(x) ≥ 0, x ∈ ℝ^{ν}. Hence H_{j.j′}^{(N)} = [G_{j,j}^{(N)}+G_{j,j′}^{(N)}+

G_{j′,j}^{(N)} + G_{j′,j′}^{(N)}] is a locally finite measure on ℝ^{ν}. Moreover H_{j,j′}^{(N)} →^{v} H_{j,j′}^{(0)} as

N → ∞. This implies that H_{j,j′}^{(N)}(Δ_{k}) → H_{j,j′}^{(0)}(Δ_{k}), therefore G_{j,j′}^{(N)}(Δ_{k}) + G_{j′,j}^{(N)}(Δ_{k}) → G_{j,j′}^{(0)}(Δ_{k}) + G_{j′,j}^{(0)}(Δ_{k}) as N → ∞ for all Δ_{k} ∈ D.

We get similarly by working with the vectors R(x|v) = (\sqrt{v(x)}, i\sqrt{v(x)}) instead of the vectors S(x|v) = (\sqrt{v(x)}, \sqrt{v(x)}) for all functions v(x) ≥ 0, x ∈ ℝ^{ν}, that K_{j.j′}^{(N)} = [G_{j,j}^{(N)} + iG_{j,j′}^{(N)} − iG_{j′,j}^{(N)} + G_{j′,j′}^{(N)}] is a a locally finite measure for all N = 0, 1, 2, …, and K_{j,j}^{(N)} →^{v} K_{j,j′}^{(0)} as N → ∞. Thus K_{j,j′}^{(N)}(Δ_{k}) → K_{j,j′}^{(0)}(Δ_{k}), therefore G_{j,j′}^{(N)}(Δ_{k}) − G_{j′,j}^{(N)}(Δ_{k}) → G_{j,j′}^{(0)}(Δ_{k}) − G_{j′,j}^{(0)}(Δ_{k}) as N → ∞ for all Δ_{k} ∈ D. These relations imply that G_{j,j′}^{(N)}(Δ_{k}) → G_{j,j′}^{(0)}(Δ_{k}) for all Δ_{k} ∈ D.

Let us define for all N = 0, 1, 2, … and our regular system D = {Δ_{k}, 1 ≤ |k| ≤ M} the Gaussian random vector

Z_{N}(D) = (Re Z_{G^{(N)},j}(Δ_{k}), Im Z_{G^{(N)},j}(Δ_{k}), |k| ≤ M, 1 ≤ j ≤ d)

I claim that the elements of the covariance matrices of the random vectors Z_{N}(D) can be expressed by means of the numbers G_{j,j′}^{(N)}(Δ_{k}), 1 ≤ |k| ≤ M and 1 ≤ j, j′ ≤ d, and the covariance matrices of Z_{N}(D) converge to the covariance matrix of Z_{0}(D) as N → ∞. (In the proof of this statement I repeat some arguments applied in the investigation of random spectral measures in Section 3 of [10].)

To prove these relations observe that

Re Z_{G^{(N)},j}(Δ_{k}) = \frac{Z_{G^{(N)},j}(Δ_{k}) + \overline{Z_{G^{(N)},j}(Δ_{k})}}{2},

Im Z_{G^{(N)},j}(Δ_{k}) = \frac{Z_{G^{(N)},j}(Δ_{k}) − \overline{Z_{G^{(N)},j}(Δ_{k})}}{2i},

and \overline{Z_{G^{(N)},j}(Δ_{k})} = Z_{G^{(N)},j}(−Δ_{k}) = Z_{G^{(N)},j}(Δ_{−k}). (In the last identity we exploited also the properties of the regular systems D.) Also the properties of the regular systems imply that if Δ_{k}, Δ_{l} ∈ D, then we have either Δ_{k} ∩Δ_{l} = Δ_{k} or Δ_{k} ∩ Δ_{l} = ∅. The first identity holds if l = k and the second one if l ≠ k. Hence we have either EZ_{G^{(N)},j}(Δ_{k})\overline{Z_{G^{(N)},j′}(Δ_{l})} = G_{j,j′}^{(N)}(Δ_{k}) if k = l or EZ_{G^{(N)},j}(Δ_{k})\overline{Z_{G^{(N)},j′}(Δ_{l})} = 0 if k ≠ l. These relations imply that we can express all covariances

ERe Z_{G^{(N)},j}(Δ_{k})Re Z_{G^{(N)},j′}(Δ_{l}), ERe Z_{G^{(N)},j}(Δ_{k})Im Z_{G^{(N)},j′}(Δ_{l})

and EIm Z_{G^{(N)},j}(Δ_{k})Im Z_{G^{(N)},j′}(Δ_{l})

with the help of the quantities G_{j,j′}^{(N)}(Δ_{k}), 1 ≤ j, j′ ≤ d, 1 ≤ |k| ≤ M. The

convergence of the numbers G_{j,j′}^{(N)}(Δ_{k}) to G_{j,j′}^{(0)}(Δ_{k}) also implies that the covariance matrices of Z_{N}(D) converge to the covariance matrix of Z_{0}(D) as N → ∞.

The convergence of the covariance matrices of the Gaussian random vectors Z_{N}(D) with expectation zero also implies that the distributions of Z_{N}(D) converge weakly to the distribution of Z_{0}(D) as N → ∞. But then the same can be told about any continuous functions of the coordinates of the random vectors Z_{N}(D). Because of the definition of the multiple Wiener–Itô integrals of simple functions the random variables Y_{N} in formula (3.18) are polynomials, hence continuous functions of the coordinates of the random vectors Z_{N}(D). Besides, these polynomials do not depend on the parameter N. Hence the previous results imply that formula (3.18) holds. Proposition 3.1 is proved. □

To simplify the application of Proposition 3.1 we also prove the following lemma.

Lemma 3.2. Let us have a sequence of matrix valued spectral measures (G_{j,j′}^{(N)}), N = 1, 2, …, 1 ≤ j, j′ ≤ d, on the torus [−A_{N}π, A_{N}π]^{ν} such that

A_{N} → ∞, and G_{j,j′}^{(N)} →^{v} G_{j,j′}^{(0)} with some complex measure (G_{j,j′}^{(0))}) with locally

finite total variation for all 1 ≤ j, j′ ≤ d as N → ∞. Then G^{(0)} = (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, is a positive semidefinite matrix valued even measure on ℝ^{ν}.

Remark. Lemma 3.2 helps to show that in many interesting cases the limit matrix G^{(0)} = (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, of the spectral measures G^{(N)} = (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, N = 1, 2, …, in Proposition 3.1 is the spectral measure of a generalized stationary Gaussian random field. Indeed, in Theorem 4.1 of [10] it was shown that a positive semidefinite matrix valued measure on ℝ^{ν} whose distribution is moderately increasing at infinity is the spectral measure of such a random field. (See Section 4 of [10] for the definition of these notions.) So by Lemma 3.2 to prove that G^{(0)} is the spectral measure of a generalized stationary Gaussian random field it is enough to show that the distribution of G^{(0)} is moderately increasing.

Proof of Lemma 3.2. We have to show that (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, is a positive semidefinite matrix valued measure. To do this take a vector v(x) = (v_{1}(x), …, v_{d}(x)) whose coordinates v_{k}(x), 1 ≤ k ≤ d, are continuous functions with compact support. We have

lim_{N→∞} ∑_{j=1}^{d} ∑_{j′=1}^{d} ∫ v_{j}(x)v_{j′}(x)G_{j,j′}^{(N)}( dx) = ∑_{j=1}^{d} ∑_{j′=1}^{d} ∫ v_{j}(x)v_{j′}(x)G_{j,j′}^{(0)}( dx) ≥ 0.

(3.19)

The identity in (3.19) holds, since G_{j,j′}^{(N)} →^{v} G_{j,j′}^{(0)} for all 1 ≤ j, j′ ≤ d. The

inequality at the end of (3.19) also holds, because (G_{j,j′}^{(N)}), 1 ≤ j, j′ ≤ d, is a positive semidefinite matrix valued measure for all N = 1, 2, …, and this implies that the left-hand side of (3.19) is non-negative for all N = 1, 2, . . .. Thus we got that if g_{j,j′}^{(0)}(x) is the Radon–Nikodym derivative of G_{j,j′}^{(0)} with respect to some dominating measure μ_{0} in the point x ∈ ℝ^{ν} for all 1 ≤ j, j′ ≤ d, we take the d × d matrix g^{(0)}(x) = (g_{j,j′}^{(0)}(x)), 1 ≤ j, j′ ≤ d, and the coordinates of the vector v(x) = (v_{1}(x), …, v_{d}(x)) are continuous functions with compact support, then

∫ v(x)g^{(0)}(x)v^{∗}(x)μ_{0}( dx) ≥ 0.

In the proof of Theorem 2.2 of [10] we have shown that this relation implies that (G_{j,j′}^{(0)}), 1 ≤ j, j′ ≤ d, is a positive semidefinite matrix valued measure.

We still have to show that the complex measure G_{j,j′}^{(0)} with locally finite variation is even for all 1 ≤ j, j′ ≤ d. To do this fix a pair j, j′ of indices, 1 ≤ j, j′ ≤ d, and define for all N = 0, 1, 2, … the complex measure (G′)_{j,j}^{(N)} by the relation (G′)_{j,j′}^{(N)}(A) = \overline{G_{j,j′}^{(N)}(−A)} for all bounded, measurable sets

A ⊂ ℝ^{ν}. It is not difficult to see that not only G_{j,j′}^{(N)} →^{v} G_{j,j′}^{(0)}, but also

(G′)_{j,j′}^{(N)} →^{v} (G′)_{j,j′}^{0} as N → ∞. The evenness of the measures G_{j,j′}^{(N)} for

N = 1, 2, … means that G_{j,j′}^{(N)} = (G′)_{j,j′}^{(N)} for all N = 1, 2, . . .. By taking the

limit N → ∞ we get that G_{j,j′}^{(0)} = (G′)_{j,j′}^{(0)}. This means that G_{j,j}^{(0)} is an even complex measure with locally finite variation. Lemma 3.2 is proved. □

Appendix A. On the results in Section 2.

I wrote an Appendix consisting of two parts. In the first part I discuss some questions related to Section 2 while in the second part some questions related to Section 3.

In the first part, I write about the idea of the proof of Itô’s formula together with the introduction of Wick polynomials. I also explain how Itô’s formula is applied in the proof of the other results in Section 2.

In the second part, I write about the investigation of limit problems for non-linear functionals a stationary Gaussian random fields and the content of Proposition 3.1. Here I want to explain the role of the compactness type condition (b) in this proposition. I shall do this with the help of an example.

First I try to explain the idea behind the proof of Itô’s formula. To do this first I consider its one-dimensional version. In that case we want to prove the following statement. Let φ(x), φ(−x) = \overline{φ(x)}, be such a function for which ∫ |φ(x)|^{2}G( dx) = 1 with a spectral measure G, and consider a random spectral measure Z_{G} corresponding to the spectral measure G. Then Y = ∫ φ(x)Z_{G}( dx) is a standard normal random variable, and the identity

H_{n} (∫ φ(x)Z_{G}( dx)) = ∫ φ(x_{1}) ⋯ φ(x_{n})Z_{G}( dx_{1}) … Z_{G}( dx_{n}). (A.1)

holds for all n ≥ 1. Actually, Itô’s formula is a more general result, but here it will be enough to consider this special case.

It is proved in the general theory that the above-defined random variable Y has standard normal distribution. Formula (A.1) is proved by induction with respect to n. In this induction, we apply the recursion formula (2.2) for Hermite polynomials and the diagram formula for the product of multiple Wiener–Itô integrals. We exploit that they “fit to each other”.

Formula (A.1) clearly holds for n = 1. To prove it for n if we know it for m < n we rewrite the left-hand side of (A.1) with the help of the recursion formula (2.2). We rewrite the random integral

∫ φ(x_{1}) ⋯ φ(x_{n})Z_{G}( dx_{1}) … Z_{G}( dx_{n})

at the right hand of (A.1) by means of the identity that we get by applying the diagram formula for the product

∫ φ(x_{1}) ⋯ φ(x_{n−1})Z_{G}( dx_{1}) … Z_{G}( dx_{n−1}) ∫ φ(x)Z_{G}( dx).

(I remark that Proposition 5.1 of [8] yields a generalization of the formula we get in such a way.) Then some calculations with the help of these formulas and the inductive hypothesis yield the proof of formula (A.1) for n.

Itô’s formula for vector-valued stationary Gaussian random fields can be proved by an appropriate adaptation of the above argument. In the proof, we apply a useful special case of the diagram formula for vector-valued stationary Gaussian random fields presented in [10]. It is formulated in the corollary of Theorem 6.1 in[10].

On the other hand, we need a new identity instead of formula (2.2) in the proof, and we can formulate it with the help of Wick polynomials. This was the reason for their introduction in this paper. They are defined in Section 2 with the help of some results in [10]. First we have to understand that this definition is correct. Namely, we have to show that the Wick polynomial : P (ξ_{1}, …, ξ_{m}): of a homogeneous polynomial P (ξ_{1}, …, ξ_{m}) of order n depends only on the random variables ξ_{1}, …, ξ_{m}, although in its definition we applied a projection to a Hilbert space H_{n} which may depend on other random variables, too.

The result of Theorem 2B implies the correctness of this definition. It states that the value of the Wick polynomial does not change if we take projection to the Hilbert space H_{n}(ξ_{1}, …, ξ_{m}) introduced before the formulation of this result instead of the projection to H_{n}. The definition of this new Hilbert space is similar to that of H_{n}, the only difference is that here we work only with the random variables ξ_{1}, …, ξ_{m}.

The proof of Theorem 2B exploits the following property of Gaussian random vectors. If some coordinates of a Gaussian random vector are uncorrelated, then they are also independent. This implies that the elements of the underlying Gaussian random field can be decomposed as X_{t} = η_{1,t} +η_{2,t}, t ∈ T, in such a way that η_{1,t} is a linear combination of the random variables ξ_{1}, …, ξ_{m}, while η_{2,t} is uncorrelated, hence independent of them. The proof of Theorem 2B is based on this fact. I omit the details of the proof.

Theorem 2B implies in particular that : ξ^{n}: = H_{n}(ξ) if ξ is a standard normal random variable. Corollary 2C describes a deeper relation between Hermite and Wick polynomials. This can be exploited. For instance, the identity (2.3) formulated in Proposition 2.1 can be proved with its help and formula (2.2) about Hermite polynomials. This identity plays an important role in the proof of Itô’s formula.

The proof of Itô’s formula for vector-valued stationary Gaussian random fields is made with the help of the identity (2.3) for Wick polynomials and the Corollary of Theorem 6.1 in [10]. (Theorem 6.1 in [10] is the diagram formula for vector-valued stationary Gaussian random fields.) It is a natural adaptation of the previously discussed proof in the scalar-valued case.

It is easy to deduce from Theorem 2.2 its Corollary. In this Corollary the Wick polynomials of such homogeneous polynomials are considered whose arguments are elements of one of the Hilbert spaces H_{1,j}, 1 ≤ j ≤ d. In the Corollary of Theorem 2.2 such expressions are expressed in the form of a sum of multiple Wiener–Itô integrals. The class of homogeneous polynomials considered in the Corollary of Theorem 2.2 is fairly large. This fact is exploited in the proof of Proposition 2.3, which states that the set of all finite sums of n-fold Wiener–Itô integrals constitute an everywhere dense class of functions in H_{n}.

In the last result of Section 2, in Theorem 2.4 a random variable, defined in formula (2.6) in the form of a multiple Wiener–Itô integral is considered, and its shift transforms are given in formula (2.8). This is an important result, and its proof is based also on the Itô formula. The validity of formula (2.8) can be checked first for the shifts of one-fold Wiener–Itô integrals. Then it can be proved with the help of the Itô formula for such multiple Wiener–Itô integrals whose kernel functions have the special form

h(x_{1}, …, x_{n}) = φ_{1}(x_{1}) ⋯ φ_{n}(x_{n}).

After this Proposition 2.4 can be proved in the general case by means of a standard method.

Appendix B. On the results in Section 3.

At the beginning of Section 3 I formulated a limit problem. I considered a vector-valued stationary Gaussian random field X(p), p ∈ ℤ^{ν}, defined a Wick polynomial of order k of the coordinates of the vector X(0) in (3.4), and I was interested in a limit problem for the expressions S_{N} defined in (3.3). These expressions are normalized partial sums whose elements are shifts T_{p}Y (0) of the random variable Y (0). This problem is a multivariate version of the problem studied in [5].

The expressions S_{N} can be rewritten in an interesting simple form with the help of Itô’s formula that enables us to rewrite Y (0) in the form of a sum of multiple Wiener–Itô integrals, and of Proposition 2.4 that gives a useful formula for its shifts T_{p}Y (0). I could express with their help and with an appropriate rescaling of the random integrals we are working with the expression S_{N} in the form given in (3.7). I applied this rescaling to get such an expression in(3.7), where the kernel functions of the multiple Wiener–Itô integrals have a limit as N → ∞. This formula suggests that if the matrix valued spectral measures G^{(N)} have a limit, then we can get the limit of the random variables S_{N} by means of a natural limiting procedure. Maybe, this can be done only if some additional not too restrictive conditions are satisfied. Proposition 3.1 gives a useful sufficient condition for the right to carry out such a limiting procedure, and to get a limit theorem in this way.

In the formulation of Proposition 3.1 some random variables Z_{N}, N = 0, 1, 2, …, defined in (3.9) and (3.12) are considered. First it is shown that under the conditions of Proposition 3.1 these random variables exist (the multiple integrals appearing in their definition are well-defined), and then also the convergence Z_{N} →^{D} Z_{0} is proved.

The first conditions of Proposition 3.1 are natural. They demand that the kernel functions h_{j_{1},...,j_{k}}^{N}(x_{1}, …, x_{k}) of the random integrals in Z_{N} and the matrix valued spectral measures G^{(N)} converge to h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k}) and G^{(0)} in an appropriate way. (Actually, there is also a hidden condition here. It is the condition that the functions h_{j_{1},...,j_{k}}^{0}(x_{1}, …, x_{k}) must be continuous. In the scalar-valued version of this result, in Proposition 8.3 of [8] a slightly weaker continuity condition is imposed. Here we do not discuss the question how the continuity property in Proposition 3.1 can be weakened.)

On the other hand, part (b) of Proposition 8.3 contains an additional condition that deserves special attention. We can understand its role better by considering the application of Proposition 3.1 in the proof of the main result in [11] or the application of its scalar-valued version in the proof of Theorem 8.2 of [8].

In Theorem 8.2 of [8] a scalar-valued stationary Gaussian random field X(p), p ∈ ℤ^{ν}, is considered. The random variables Y (p) = H_{k}(X(p)) = T_{p}Y (0) are introduced, where H_{k}(·) is the Hermite polynomial of order k, and the normalized partial sums S_{N} are defined by formula (3.3) with these random variables Y (p). A non-central limit theorem is proved with normalizing constants A_{N} = N^{ν−kα/2}L(N)^{k/2} if the corelation function r(p) = EX(0)X(p) satisfies the relation r(p) = |p|^{−α}a (\frac{p}{|p|}) L(|p|) with some 0 < α < \frac{ν}{k}, where L(·) is a slowly varying function at infinity, and a(·) is a function on the unit sphere of ℝ^{ν}. It shows the dependence of the correlation function r(p) on the direction of the vector p.

Paper [11] contains a multivariate version of this result. Here a vectorvalued stationary Gaussian random field X(p) = (X_{1}(p), …, X_{d}(p)), p ∈ ℤ^{ν}, is considered, and a limit theorem is proved for the random variables S_{N}, defined in (3.3) and (3.4) under appropriate condition. A condition, similar to the condition of the correnlation function in Theorem 8.2 of [8] is imposed on the correlation function r_{j,j′}(p) = EX_{j}(0)X_{j′}(p) which is described in formula (1.3) of [10]. A non-central limit theorem with normalizing constants A_{N} = N^{ν−kα/2}L(N)^{k/2} is proved if this condition holds for the correlation function with exponent 0 < α < \frac{ν}{k}.

It is worth understanding why the condition α < \frac{ν}{k} in the exponent of the formula expressing the decrease of the correlation function is needed in the proof of these results. In the first step of these proofs, we have to describe the asymptotic behavior of the spectral measure of the underlying stationary Gaussian random field. In the scalar-valued this is done in Lemma 8.2 of [8]. It describes the limit behavior of the appropriately rescaled versions G_{N} of the spectral measure G of the stationary Gaussian random field we are working with. These measures are defined by the identity G_{N}(A) = \frac{N^{α}}{L(N)}G (\frac{A}{N} ) for all measurable sets A. In this lemma, it is proved that the measures G_{N} have a vague limit G_{0} as N → ∞, and the limit measure has the homogeneity property G_{0}(A) = t^{−α}G_{0}(tA) for all measurable sets A and t > 0. There is a similar result also in the case of vector-valued stationary Gaussian fields.

The above results imply that the spectral measures and kernel functions in the representation of S_{N} in formula (3.7) satisfy the starting conditions of Proposition 3.1 if the correlation function of the underlying stationary Gaussian field has such an asymptotic behavior at the infinity as it is demanded in the above-mentioned results. Moreover, they are satisfied if the correlation function satisfies the desired relation with arbitrary “homogeneity parameter” ν > α > 0.

One would like to understand where we exploited the condition α < \frac{ν}{k}, and what can be told if this condition is violated. (The number k was the order of the Wick polynomial we were working with.) This question is related to condition (b) in Proposition 3.1. It is proved that this condition is satisfied under the additional condition α < \frac{ν}{k}. (This is proved in [5] or [8] in the scalar and in [11] in the vector-valued case.) On the other hand, it can be proved that if the asymptotic formula prescribed for the correlation function satisfies the prescribed asymptotic relation with α ≥ \frac{ν}{k}, then relation (3.11) does not hold, and as a consequence, the definition of the random variable Z_{0} in (3.12) is incorrect.

We have more detailed knowledge of the behavior of the random sum S_{N} defined in (3.3) if α > \frac{ν}{k}. It is known that it satisfies the central limit theorem with the standard normalization N^{ν/2}. This follows from the central limit theorem proved in [3] in the scalar and in Theorem 4 of [1] in the vectorvalued case. One only has to check that the conditions of these results are satisfied in this case, and this can be done by calculating the necessary covariances. A similar central limit theorem also holds if α = \frac{ν}{k}, but in this case, it may happen that the norming constant is N^{ν/2}L(N) with a slowly varying function L(N) tending to infinity as N → ∞.

The above-discussed results suggest the following heuristic picture about a generalized version of the results discussed in the Appendix.

Let us have a stationary Gaussian random field X(p), p ∈ ℤ^{ν} and a non-linear functional Y of this random field. Take the shifts Y (p) = T_{p}Y, p ∈ ℤ^{ν}, of this non-linear functional, and consider their normalized sums S_{N} defined in formula (3.1). We are interested in what kind of limit theorem holds for this sequence S_{N} as N → ∞ if we choose the norming constants A_{N} appropriately. In Section 3 we considered a special case of this problem and proved by applying a method that it can be reformulated to the problem about the limit theorem for a sequence Z_{N} defined in (3.9). Such a sequence has a limit if the kernel functions and spectral measures in the definition of Z_{N} behave nicely. A similar reformulation of the above-mentioned limit problem is possible in a more general case, only different kernel functions appear in the definition of the random integrals in (3.9). Proposition 3.1 may help in the study of the limit behavior of the random variables Z_{N} defined in (3.9). It turned out that condition (b) of Proposition 3.1 is an important condition of this result. Let us understand its role better.

Condition (b) of Proposition 3.1 is a compactness type condition formulated in (3.10). Let us consider the integrals in it when we integrate on the whole space ℝ^{ν}. If the values of these integrals tend to infinity as N → ∞, then for large N the essential part of the random integrals in (3.9) comes from a region that contains vectors with a very big norm in ℝ^{kν}. It is natural to expect that in such cases the random variables Z_{N} satisfy the central limit theorem with the classical norming constants A_{N} = N^{ν/2} under very general conditions. Such a result is proved beside the above-mentioned papers [1] and [3] also in the book [12]. On the other hand, Proposition 3.1 implies the existence of a non-Gaussian limit, expressed by means of a sum of multiple Wiener–Itô if the the spectral measures and the kernel functions have a limit, and condition (b) of Proposition 3.1 holds.

Acknowledgements. This research was supported by the Hungarian Foundation NKFI–EPR No. K-125569.

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[10] Major. P. (2020) The theory of Wiener–Itô integrals in vector-valued Gaussian random fields. Part I. Moscow Math. Journal, Volume 20, Number 4, 1–65.

[11] Major, P. (2021) Non-central limit theorem for non-linear functionals of vector-valued Gaussian stationary random fields. To be submitted to the Annals of Probability

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Alfréd Rényi Institute of Mathematics Budapest, P.O.B. 127 H–1364, Hungary

Email address: major@renyi.hu