ESTIMATION OF WIENER–ITÔ INTEGRALS AND OF POLYNOMIALS OF INDEPENDENT GAUSSIAN RANDOM VARIABLES.

Péter Major Alfréd Rényi Mathematical Institute of the Hungarian Academy of Sciences

In this paper I prove good estimates on the moments and tail distribution of k-fold Wiener–Itô integrals and also present their natural counterpart for polynomials of independent Gaussian random variables. The proof is based on the so-called diagram formula for Wiener–Itô integrals which yields a good representation for their products as a sum of such integrals. In a subsequent paper I shall show that this method also enables us to prove a natural counterpart of these estimates for U-statistics. The main result of this paper is a generalization of the estimates of Hanson and Wright about bilinear forms of independent standard normal random variables. On the other hand, it is a weaker estimate than the main result of a paper of Latała [5]. But that paper contains an error, and it is not clear whether its result is true. This question is also discussed here.

1. Introduction. Formulation of the main results.

The goal of this paper is to give good estimates on the tail-distribution and on high moments of Wiener–Itô integrals. This problem can be reformulated in an equivalent form to the estimation of polynomials of independent Gaussian random variables. The results obtained in such a way will be also proved.

This paper can be considered as a continuation of my investigation in paper [9] where good estimates were given for the tail-distribution and high moments of Wiener– Itô integrals and (degenerate) U-statistics with the help of their variance. The results of [9] were proved by means of the so-called diagram formula which yields a useful expression for the moments of Wiener–Itô integrals or degenerate U-statistics. In the present paper it is shown that this method can be applied also in cases when we have more information about a Wiener–Itô integral than its variance and want to exploit this. In a subsequent paper I shall show in a subsequent paper [10] that similar improvements can be proved about the moments and tail-distribution of degenerate U-statistics.

Previous papers in this field (see [3] or [5]) dealt with the estimation of polynomials of independent standard normal random variables and not of Wiener–Itô integrals. I shall show that the result of [3] about the estimation of bilinear forms of independent standard normal random variables is equivalent to the special case of the main result in this paper when only two-fold Wiener–Itô integrals are considered. On the other hand, paper [5] formulates sharper estimates about Gaussian polynomials of higher order than our results. But the proof in [5] contains an error, hence some problems arise with respect to this paper. I return to this question later.

First I introduce some notations needed to formulate the results of the present paper.

Let us have a σ-finite non-atomic measure μ on some measurable space (X, X) together with a white noise μ_{W} with reference measure μ, i.e. a set of jointly Gaussian random variables μ_{W} (B) indexed by the sets B ∈ X such that μ(B) < ∞, whose joint distribution is determined by the relations Eμ_{W} (B) = 0, Eμ_{W} (A)μ_{W} (B) = μ(A ∩ B) for all sets A, B ∈ X such that μ(A) < ∞, μ(B) < ∞. Let us also introduce the quantity

V_{1}^{2}(f) = ∫ f^{2}(x_{1}, …, x_{k})μ( dx_{1}) … μ( dx_{k})  (1.1)

for a function f of k variables on the space (X, X). The k-fold Wiener–Itô integral

I_{k}(f) = \frac{1}{k!} ∫ f(x_{1}, …, x_{k})μ_{W} ( dx_{1}) … μ_{W} ( dx_{k})  (1.2)

can be defined for all functions f such that V_{1}(f) < ∞. (See e.g. [4] or [7].) Here the knowledge of the definition of Wiener–Itô integrals is not assumed. We only need the so-called the diagram formula which enables us to calculate the moments of these random integrals. This result will be recalled in Section 3.

We are interested in good estimates on the probability P (k!|I_{k}(f)| > x) for large numbers x > 0. This problem is closely related to the question about good moment estimates E(k!I_{k}(f))^{2M} for large values M. The results of paper [9] yield a good estimate for these quantities with the help of V_{1}(f). I recall them in the following Theorem A.

Theorem A. Let a measurable space (X, X) be given together with a non-atomic σfinite measure on it. Let μ_{W} be a white-noise with reference measure μ, and let such a function f(x_{1}, …, x_{k}) of k variables be given on the space (X, X) for which V_{1}^{2}(f) ≤ σ^{2} with some σ < ∞, where the quantity V_{1}(f) was defined in (1.1). Then the k-fold Wiener–Itô integral I_{k}(f) introduced in (1.2) satisfies the inequalities

E(k!I_{k}(f))^{2M} ≤ C (\frac{2kM}{e} )^{kM} σ^{2M} for all M = 1, 2, …  (1.3)

and

P (k!|I_{k}(f)| > x) ≤ C exp \{−\frac{1}{2} (\frac{x}{σ} )^{2/k}\} for all x > 0  (1.4)

with some appropriate universal C > 0 depending only on the multiplicity k of the Wiener–Itô integral.

This estimate is sharp in the following sense. There are functions f(x_{1}, …, x_{k}), (a function of the form f(x_{1}, …, x_{k}) = g(x_{1}) … g(x_{k}), ∫ g^{2}(x)μ( dx) < ∞, is an appropriate choice) for which the constant in the exponent of the probability estimate (1.4) cannot be increased, i.e. the inequality P (k!|I_{k}(f)| > x) ≤ Ce^{−K(x/σ)^{2/k}} does not hold for K > \frac{1}{2}.

Let me remark that EI_{k}^{2}(f) = \frac{1}{k!}V_{1}^{2}(f), and EI_{k}(f) = 0. So Theorem A gave a good estimate on the moments and tail-distribution of Wiener–Itô integrals with the help of their variance. If we have no more information about the kernel function f of the Wiener–Itô integral I_{k}(f), than a bound on V_{1}(f) then we cannot improve the estimates in Theorem A. On the other hand, one may introduce new quantities to get better estimates. In this paper such results will be proved. For this goal this first I introduce some new quantities.

Let us fix a function f(x_{1}, …, x_{k}) on (X, X), introduce the set K = {1, …, k} of the indices of its variables, and let P = P(K) denote the set of partitions of the set K to non-empty sets. Given a P = {A_{1}, …, A_{s}} ∈ P(K) let us define the class of appropriate sequences of functions denoted by F_{P} by the following formula:

\begin{aligned}F_{P} & = \{g_{r}(x_{j}, j ∈ A_{r}), 1 ≤ r ≤ s: \\ & ∫ g_{r}^{2}(x_{j}, j ∈ A_{r}) ∏_{j∈A_{r}} μ(dx_{j}) ≤ 1 for all 1 ≤ r ≤ s\} \end{aligned}  (1.5)

if P = {A_{1}, …, A_{s}} ∈ P(K). This means that F_{P} consists of a sequence of functions g_{r} on (X, X) whose variables are indexed by the elements of corresponding sets A_{r} in the partition P, and with L_{2}-norm less than or equal to 1 with respect to the appropriate product of the copies of the measure μ.

Let us define for all sets P = {A_{1}, …, A_{s}} ∈ P the quantity

V_{P} (f) = sup_{(g_{1},...,g_{s})∈F_{P}} ∫ f(x_{1}, …, x_{k}) ∏_{1≤r≤s} g_{r}(x_{j}, j ∈ A_{r}) ∏_{j∈K} μ( dx_{j}).  (1.6)

Beside this, introduce the class P_{s} = P_{s}(K) ⊂ P(K) of partitions of K which consist of exactly s elements, and put

V_{s}(f) = sup_{P ∈P_{s}} V_{P} (f).  (1.7)

Let me remark that the definition of V_{1}(f) introduced in (1.1) agrees with the definition of V_{s}(f) in (1.7) for s = 1. Beside this, the definition of the quantities F_{P}, V_{P} (f) and V_{s}(f) were formulated in such a way that they are meaningful for any set K = {u_{1}, …, u_{t}} and square integrable function f(x_{u_{1}}, …, x_{u_{t}}) with variables indexed by the set K. In the formulation of our results we can restrict our attention to the case K = {1, …, k}, but in later considerations we shall also work with these quantities in the case of more general sets K.

The main result of this paper, an estimate about the moments and tail distribution of a Wiener–Itô integral I_{k}(f), can be formulated with the help of the quantities V_{s}(f) introduced in formulas (1.5), (1.6) and (1.7).

Theorem about the tail-distribution of Wiener–Itô integrals. Let us consider a k-fold Wiener–Itô integral I_{k}(f), k ≥ 2, defined in (1.1) by means of a white noise μ_{W} with a non-atomic σ-additive reference measure μ on a measurable space (X, X) and a measurable function f(x_{1}, …, x_{k}) of k variables on the space (X, X) such that V_{1}^{2}(f) < ∞ for the quantity V_{1}(f) defined in (1.1). Then there exist some universal constants C > 0, C_{1} > 0 and C_{2} > 0 depending only on the multiplicity k of the Wiener–Itô integral I_{k}(f) such that

E(k!I_{k}(f))^{2M} ≤ C^{M} max_{1≤s≤k} (V_{s}(f)^{2M}M^{[(s−1)(k−1)+1]M}) , for all M = 1, 2, …, (1.8)

and

P (|k!I_{k}(f)| > x) ≤ C_{1} exp \{−C_{2} min_{1≤s≤k} (\frac{x}{V_{s}(f)})^{2/[(s−1)(k−1)+1]}\}  (1.9)

with the quantities V_{s}(f), 1 ≤ s ≤ k, defined in formulas (1.5), (1.6) and (1.7) with the set K = {1, …, k}.

It will be more convenient to prove first the following simpler version of this theorem and to deduce the result in the general case from it.

Simplified version of the theorem about the tail distribution of Wiener–Itô integrals. Let a k-fold Wiener–Itô integral I_{k}(f), k ≥ 2, be considered with respect to a white noise μ_{W} with reference measure μ whose kernel function f satisfies the inequalities

V_{s}(f) ≤ M^{−(s−1)(k−1)/2} for all 1 ≤ s ≤ k  (1.10)

with some positive integer M. Then the inequality

E(k!I_{k}(f))^{2M} ≤ C^{M}M^{M}  (1.11)

holds with some universal constant C depending only on the multiplicity k of the Wiener– Itô integral.

Reduction of the theorem about the tail-distribution of Wiener–Itô integrals to its simplified version. Let us introduce with the help of the kernel function f of a Wiener–Itô integral I_{k}(f) the number B(M) = max_{1≤s≤k} V_{s}(f)M^{(k−1)(s−1)/2}), and put f̄ = f̄(M) =

\frac{1}{B(M)}f. Then the function f̄ satisfies the conditions of the reduced theorem, hence it implies that relation (1.11) holds for the Wiener–Itô integral with kernel function f̄. This relation implies formula (1.8). Formula (1.9) can be proved in the standard way by means of formula (1.8) and the Markov inequality P (|k!I_{k}(f)| > x) ≤ \frac{E(k!I_{k}(f))^{2M}}{x^{2M}} with a good choice of the parameter M. The choice of the closest integer to

C̃ min_{1≤s≤k} (\frac{x}{V_{s}(f)})^{2/[(s−1)(k−1)+1]}

for the parameter M with a sufficiently small C̃ > 0 (if x > 0 is sufficiently large) supplies formula (1.9).

This paper contains the proof of the above result. But before turning to it I discuss what kind of estimates it yields for polynomials of independent standard normal random variables. This will be the subject of Section 2. Beside this, this section contains a comparison of these estimates with earlier results in this field. In particular, a result of Latała is discussed there together with the problem appearing in its proof. Section 3 contains a formula which helps to calculate the moments of a Wiener–Itô integral. This formula is a consequence of the diagram formula for the product of Wiener–Itô integrals, and it enables us to prove good moment estimates for Wiener–Itô integrals if we can find good bounds on some integrals defined with the help of some diagrams. A closer study showed that it is useful to consider a special class of the diagrams, the so-called connected diagrams, and it is enough to estimate the integrals related to them. A result about a good bound on such integrals, called the Basic Estimate is also formulated in Section 3, and it is shown that the proof of the main result of this paper can be reduced to that of the Basic Estimate. In Section 4 a result called the Main Inequality is proved, and the Basic Estimate is proved with its help. Finally in Section 5 Latała’s result about an improvement of our estimates is discussed in more detail, and the question is investigated what kind of result has to be solved to decide whether it is true.

2. Bounds on random polynomials of Gaussian random variables.

The natural counterpart of the estimation of k-fold Wiener–Itô integrals is the estimation of some special polynomials of order k of independent standard normal random variables defined with the help of Hermite polynomials. In more detail, polynomials of the following form will be considered. Let us take independent, standard normal random variables ξ_{1}, ξ_{2} …, and introduce with their help the following random polynomials.

Z_{k} = ∑_{\begin{matrix}((j_{1},l_{1}),...,(j_{s},l_{s})), \\ j_{u}≠j_{u′} if u≠u′, l_{1}+···+l_{s}=k \end{matrix}} a((j_{1}, l_{1}), …, (j_{u}, l_{u}))H_{l_{1}}(ξ_{j_{1}}) ⋯ H_{l_{s}}(ξ_{j_{s}}).  (2.1)

Here the coefficients a((j_{1}, l_{1}), …, (j_{u}, l_{u})) are some real numbers, and H_{l}(x) denotes the Hermite polynomial of order l with leading coefficient 1. For the sake of simplicity let us assume that the sum in formula (2.1) contains only finitely many terms. Infinite sums could also be allowed in formula (2.1), but in such a case one should deal with some convergence problems.

It is more convenient to rewrite the random polynomials Z_{k} in formula (2.1) in a different form. This new notation introduced below means to work with Wick polynomials of Gaussian random variables, i.e. to apply a multivariate generalization of Hermite polynomials. To introduce the new notation put

H_{l_{1}}(ξ_{j_{1}}) ⋯ H_{l_{s}}(ξ_{j_{s}}) =:\underbrace{ξ_{j_{1}}, …, ξ_{j_{1}}}_{l_{1}-times}, …, \underbrace{ξ_{j_{s}}, …, ξj_{s}}_{l_{s}-times}: .

At the right-hand side of the last formula there is the product of k Gaussian random variables ξ_{j_{s}}, 1 ≤ j_{s} ≤ n, 1 ≤ s ≤ k, between the two signs :, and some of the terms ξ_{j_{s}} may agree. For the sake of a more convenient notation let us slightly extend the definition of the above expression. Let us allow to write the terms in this product in an arbitrary order, i.e. put

:ξ_{j_{π(1)}}, …, ξ_{j_{π(k)}} :=:ξ_{j_{1}}, …, ξ_{j_{k}}:

for all permutations π = (π(1), …, π(k)) of the set {1, …, k}. With such a notation the random polynomials we are working with can be rewritten in the form

Z_{k} = ∑ a(n_{1}, …, n_{k}) :ξ_{n_{1}}ξ_{n_{2}} ⋯ ξ_{n_{k}}:  (2.2)

with some real coefficients a(n_{1}, …, n_{k}) that can be calculated by means of the coefficients a((j_{1}, l_{1}), …, (j_{u}, l_{u})) in formula (2.1). In the subsequent considerations I shall estimate random polynomials presented in the form (2.2).

Let us consider the unit interval [0, 1] together with the Lebesgue measure on it denoted by μ, and let μ_{W} be a white noise with this reference measure μ. Take a complete orthonormal system of functions φ_{1}(x), φ_{2}(x), … with respect to the Lebesgue measure on [0, 1] and put ξ_{n} = ∫ φ_{n}(x)μ_{W} ( dx), n = 1, 2, . . .. Then ξ_{1}, ξ_{2}, … is a sequence of independent, standard normal random variables, and we do not change the distribution of the random polynomial Z_{k} in (2.2) by choosing the above introduced standard normal random variables ξ_{n} in it. Thus we may assume that these standard normal random variables appear in the definition of the random polynomial Z_{k}, and I shall exploit this liberty. With such a choice Itô’s formula for multiple Wiener–Itô integrals (see e.g. [4]) enables us to rewrite the random polynomial Z_{k} in the form of a k-fold Wiener–Itô integral. Such a representation of the random polynomial Z_{k} is useful, because it enables us to apply the results we know about Wiener–Itô integrals in our investigations.

To find the above indicated representation observe that by Itô’s formula

:ξ_{n_{1}}ξ_{n_{2}} ⋯ ξ_{n_{k}}:= ∫ φ_{n_{1}}(x_{1}) ⋯ φ_{n_{k}}(x_{k})μ_{W} ( dx_{1}) … μ_{W} ( dx_{k}),

hence

Z_{k} = k!I_{k}(f) = ∫ f(x_{1}, …, x_{k})μ_{W} ( dx_{1}) … μ_{W} ( dx_{k})  (2.3)

with

f(x_{1}, …, x_{k}) = ∑ a(n_{1}, …, n_{k})φ_{n_{1}}(x_{1}) ⋯ φ_{n_{k}}(x_{k}).  (2.4)

We get an estimate for the moments and tail-distribution of the random polynomial Z_{k} with the help of the results formulated in Section 1 if we can express the quantities V_{s}(f), 1 ≤ s ≤ k, for the function f introduced in (2.4) by means of the coefficients a(n_{1}, …, n_{k}) in formula (2.2). We shall define some quantities V̄_{s}, 1 ≤ s ≤ k, with the help of the coefficients a(n_{1}, …, n_{k}) and show that they are equal to V_{s}(f), 1 ≤ s ≤ k, with the function f defined in (2.4).

To define the desired quantities let us first introduce the set G_{P} corresponding to F_{P} defined in (1.5) for a partition P = {A_{1}, …, A_{s}} ∈ P(K) of the set K = {1, …, k}.

G_{P} = {b_{r}(n_{j}, j ∈ A_{r}), 1 ≤ n_{j} < ∞ for all j ∈ A_{r}, 1 ≤ r ≤ s:

∑_{1≤n_{j}<∞, 1≤j≤r} b_{r}^{2}(n_{j}, j ∈ A_{r}) ≤ 1 for all 1 ≤ r ≤ s}

if P = {A_{1}, …, A_{s}} ∈ P(K). With the help of this notion define similarly to the definition of V_{P} in (1.6) the quantity

V̄_{P} (a(·)) = sup_{(b_{1},...,b_{s})∈G_{P}} ∑ a(n_{1}, …, n_{k}) ∏_{1≤r≤s} b_{r}(n_{j}, j ∈ A_{r})  (2.5)

for a partition P = {A_{1}, …, A_{s}} ∈ P(K) and a sequence a(n_{1}, …, n_{k}), 0 ≤ n_{j} < ∞ for all 1 ≤ j ≤ k such that ∑ a^{2}(n_{1}, …, n_{k}) < ∞. Finally, put, similarly to formula (1.7)

V̄_{s}(a(·)) = sup_{P ∈P_{s}} V̄_{P} (a(·)).  (2.6)

I claim that

V̄_{P} (a(·)) = V_{P} (f)  (2.7)

for all partitions P ∈ P(K), and as a consequence

V̄_{s}(a(·)) = V_{s}(f)  (2.8)

with the function f defined in (2.4) and the sequence a(n_{1}, …, n_{k}) appearing in formula (2.2).

To show that relation (2.7) holds fix some partition P = {A_{1}, …, A_{s}} ∈ P(K) with A_{r} = (j_{1}^{(r)}, …, j_{l(r)}^{(r)}), 1 ≤ r ≤ s, take a set of sequences

(b_{1}(n_{j}, j ∈ A_{1}), …, b_{s}(n_{j}, j ∈ A_{s})) ∈ G_{P} ,

and correspond to it the set of functions (g_{1}, …, g_{r}) defined by the formula

g_{r}(x_{j}, j ∈ A_{r}) = ∑_{n_{j}^{(u)}: 1≤n_{j{(r) \atop u}} <∞, 1≤u≤l(r)}b_{r}(n_{j_{1}^{(r)}}, …, n_{j_{l(r)}^{(r)}}) ∏_{u=1}^{l(r)} φ_{n_{j{(r) \atop u}}} (x_{j_{u}^{(r)}}), 1 ≤ r ≤ s.

(2.9)

Then it follows from the Parseval formula that

∫ g_{r}^{2}(x_{j}, j ∈ A_{r}) ∏_{j∈A_{r}} μ( dx_{j}) = ∑_{n_{j}: 1≤n_{j}<∞, j∈A_{r}} b_{r}^{2}(n_{j}, j ∈ A_{r}), 1 ≤ r ≤ s,

(g_{1}, …, g_{r}) ∈ F_{P}, and the mapping b_{r} → g_{r}, 1 ≤ r ≤ s, defined in formula (2.9) is a one-to-one map from G_{P} to F_{P}. Beside this, the Parseval formula also implies that

∑ a(n_{1}, …, n_{k}) ∏_{r=1}^{s} b_{r}(n_{j}, j ∈ A_{r}) = ∫ f(x_{1}, …, x_{k}) ∏_{r=1}^{s} g_{r}(x_{j}, j ∈ A_{r}) ∏_{j=1}^{k} μ( dx_{j})

(2.10)

with the sequences a(n_{1}, …, n_{k}) in (2.2) and the function f(x_{1}, …, x_{k}) in (2.4).

Then taking the supremum of both sides of formula (2.10) for all (b_{1}, …, b_{r}) ∈ G_{P}, and esploiting that the two suprema equal we get relation (2.5). Indeed, the supremum of the left-hand side equals V̄_{P} ((a(·) by definition. The supremum of the right-hand side equals V_{P} (f), because as the properties of the above defined one-to-one mapping imply that the supremum of the right-hand side equals the supremum of the same expressions if supremum is taken for all (g_{1}, …, g_{s}) ∈ F_{P}. Finally, taking supremum in formula (2.5) for all P ∈ P_{s} we get formula (2.6).

The above considerations together with the theorem about the tail-distribution of Wiener–Itô integrals formulated in Section 1 supply the following result.

Theorem about the estimation of moments and tail distribution of polynomials of independent standard normal random variables. Let us consider the random polynomial Z_{k} of standard normal random variables defined in formula (2.2) There exist some universal constants C > 0, C_{1} > 0 and C_{2} > 0 depending only on the order k of this random polynomial such that

EZ_{k}^{2M} ≤ C^{M} max_{1≤s≤k} (V̄_{s}(a(·))^{2M}M^{[(s−1)(k−1)+1]M}) , for all M = 1, 2, …,

and

P (|Z_{k}| > x) ≤ C_{1} exp \{−C_{2} min_{1≤s≤k} (\frac{x}{V̄_{s}(a(·))})^{2/[(s−1)(k−1)+1]}\}  (2.11)

with the quantities V̄_{s}(f), 1 ≤ s ≤ k, defined in formulas (2.5) and (2.6).

The paper of Hanson and Wright [3] contains some estimates on the tail distribution of a bilinear form

S_{n} = ∑_{i,j=1}^{n} a(i, j)(ξ_{i}ξ_{j} − Eξ_{i}ξ_{j}),  (2.12)

where ξ_{1}, …, ξ_{n} are independent standard normal random variables, and A = (a(i, j)), 1 ≤ i, j ≤ n, is an n × n symmetric matrix. Hanson and Wright introduced the Hilbert–

Schmidt norm Λ^{2} = ∑_{i,j=1}^{n} a(i, j)^{2} and the usual norm ∥A∥ = sup_{|x|=1} |Ax| of the matrix

A = (a(i, j)), where |x| denotes the usual Euclidean norm of the vector x = (x_{1}, …, x_{n}). They proved the following estimate for the tail distribution of S_{n}.

P (S_{n} > x) ≤ C_{1} exp \{− min \frac{C_{2}x}{∥A∥}, \frac{C_{2}x^{2}}{Λ^{2}} \}  (2.13)

for all x > 0 with some universal constants C_{1} > 0 and C_{2} > 0.

This inequality agrees with the estimate (2.11) in the theorem about the estimation of moments and tail distribution of polynomials of independent standard normal random variables in the case k = 2. Indeed, the random polynomials Z_{k} defined in (2.2) agree with the polynomials S_{n} in (2.12), if they contain the same coefficients a(i, j). Beside this, in this case V̄_{1}((a(·)^{2} = Λ^{2}, and V̄_{2}(a(·) = ∥A∥ for the terms V̄_{1}(a(·)) and V̄_{2}(a(·)) defined in (2.5) and (2.6). The condition that the matrix A = ((a(i, j)) must be symmetric does not mean a real restriction, because a general polynomial S_{n} can be replaced by its symmetrization, which does not change its distribution. With such a notation the estimate (2.13) agrees with the estimate (2.11) for k = 2.

Latała (see [5]) studied the estimation of polynomials of independent standard normal random variables which have the following special form.

Z_{k} = ∑_{1≤j_{s}≤n, 1≤s≤k} a(j_{1}, …, j_{k})ξ_{j_{1}}^{(1)} … ξ_{j_{k}}^{(k)},  (2.14)

where ξ_{1}^{(s)}, …, ξ_{n}^{(s)}, 1 ≤ s ≤ k, are independent random sequences of independent standard normal random variables. In this case he formulated an estimate sharper than ours. But the proof of this result contains an error, and it is not clear whether Latała’s estimate holds. Hence I present it as a conjecture. Its formulation presented here slightly is slightly different from that of paper [5], but they are equivalent. They have a similar relation to each other as the original and simplified versions of the theorem about the tail distribution of Wiener–Itô integrals in Section 1.

Latała’s conjecture. Let the coefficients of the random Gaussian polynomial Z_{k} of order k defined in (2.14) satisfy the inequality V̄_{s}(a(·)) ≤ M^{−(s−1)/2} with some positive integer M for all 1 ≤ s ≤ k, with the quantities V̄_{s}(a(·)) defined in formulas (2.5) and (2.6). Then there exists some universal constant C depending only on the order k of the random polynomial Z_{k} such that

EZ_{k}^{2M} ≤ C^{M}M^{M}.  (2.15)

It follows from a result of de la Peña and Montgomery–Smith [1] that if Latała’s conjecture holds for random polynomials of the form (2.14), then it also holds for polynomials of the form

Z_{k} = ∑_{1≤j_{s}≤n, 1≤s≤k, j_{s}≠j_{s′} if s≠s′} a(j_{1}, …, j_{k})ξ_{j_{1}} … ξ_{j_{k}},

where ξ_{1}, …, ξ_{n} are independent standard normal random variables. With the help of this observation and some additional work Latała’s conjecture can be verified for general polynomials of the form (2.2), provided that it holds in its original form. I omit the details.

On the other hand, Latałas argument heavily exploited the special form of the random polynomials in (2.14). In the study of these polynomials it was important that the terms in the sum (2.14) contain such products ξ_{j_{1}}^{(1)} … ξ_{j_{k}}^{(k)} whose terms are elements of independent copies of a random sequences. This fact enabled Latała to apply a conditioning argument, and he could reduce the original problem to the estimation of the supremum of an appropriately defined class of Gaussian random variables with its help. But the estimation of such a supremum is very hard, and at this point some problems arise about Latała’s proof. A problem of the following type has to be considered.

Let us have a set of (jointly) Gaussian random variables, η(x), Eη(x) = 0, Eη^{2}(x) ≤ 1, x ∈ X, indexed by a parameter set X, and try to give a good estimate on the expres-

sion E (sup_{x∈X} η(x))^{2M} for large positive integers M. To study this problem let us in-

troduce the (pseudo)metric ρ_{α} defined by the formula ρ_{α}(x, y) = (E(η(x) − η(y))^{2})^{1/2}, x, y ∈ X, in the parameter space X. There is a natural way to give good estimates on the moments of the supremum we are interested in if we can give for all ε > 0 a good estimate on the minimal number N(X, ρ_{α}, ε) of balls of radius ε with respect to the distance ρ_{α} which cover the whole parameter set X.

The value N(X, ρ_{α}, ε) can be estimated in the following special case. If a probability measure μ_{ε} can be introduced in the parameter set X (or on an extension of the set X as it is done in Latała’s paper) for all ε > 0 in such a way that the estimate μ_{ε}(y: ρ(x, y) > ε/2) ≥ H(ε) is valid with some function H(·) for all x ∈ X, then the inequality N(X, ρ_{α}, ε) ≤ \frac{1}{H(ε)} holds. Latała claimed that such a construction is possible in the problem he is investigating. He formulated two lemmas, Lemma 1 and Lemma 2 in his paper which supply a good estimate, presented in Corollary 2, on N(X, ρ_{α}, ε).

However, the proof of these Lemmas 1 and 2 is problematic. Lemma 1 contains a small inaccuracy (it states the upper bound e^{−t^{2}/2} instead of e^{−1/2t^{2}}, and this wrong formula is written rather consequently), but this seems to be a corrigible error. The main problem is that Lemma 1 supplies a much weaker estimate than Lemma 2 which is not sufficient to get the results of the paper. At the explanation of this point I refer to the notation of the paper [5].

The right formula in the first line of the proof of Lemma 2 for d = 1 (page 2319 in paper [5]) would be

B_{α}(x, W_{d}^{[x]}(α, 4t) = {y ∈ R^{n_{1}}: α(x − y) ≤ W_{d}^{[x]}(α, 4t)}
= {y ∈ R^{n_{1}}: α(x − y) ≤ 4tα(xG_{n_{1}})},

where xG_{n_{1}} = (x_{j}g_{j}, 1 ≤ j ≤ n_{1}), with a standard normal random vector G_{n_{1}} = (g_{1}, …, g_{n_{1}}). Hence in the proof of Lemma 2 with d = 1 Lemma 1 should be applied with the parameter t̄ determined by the equation 4tα(G_{n_{1}}) = 4t̄α(xG_{n_{1}}) (instead of the parameter t as it is done in [5]). This number t̄ can be very small, since we have to

consider such vectors x for which ∑_{j=1}^{n_{1}} x_{j}^{2} ≤ 1. Hence Lemma 1 does not supply a good

estimate in such a case. In particular, the estimate we get, depends on the dimension n_{1} of the space R^{n_{1}}, i.e. of the space where the random vector G_{n_{1}} takes its values. On the other hand, we need such estimates which do not depend on this dimension.

The question arises whether the proof can be saved despite of this error. I think that the hardest problem about Latała’s proof is hiding behind this question. In Section 5 I return to it. For the sake of simpler notations I shall consider only the case k = 3. I show that Latała’s conjecture is equivalent to a rather hard estimate on the expected value of the supremum of some random multilinear forms whose study demands new ideas.

3. The diagram formula for Wiener–Itô integrals.

In this section the diagram formula for products of Wiener–Itô integrals is formulated, and it is shown how the proof of the simplified version of the theorem about the tail distribution of Wiener–Itô integrals can be reduced with its help to an estimate that I call the Basic Estimate.

The diagram formula makes possible to rewrite the product of Wiener–Itô integrals as a sums of such integrals, and as a consequence, it supplies a formula about the moments of Wiener–Itô integrals. It was shown in [8] that this formula yields a good estimate on the moments of Wiener–Itô integrals. In this paper it will be shown that if the quantities V_{s}(f), 1 ≤ s ≤ k, defined in (1.6) and (1.7) are very small, then this method yields a better moment estimate. I recall this formula. It is the same result that I presented in paper [8], only I made some small changes in the notation. The indices of the arguments of the functions we are working with will be indexed in a different way, because this simplifies the discussion.

The following problem is considered: Let us have m such real-valued functions

f_{j}(x_{(j,1)}, …, x_{(j,k_{j})}), 1 ≤ j ≤ m

with k_{j} variables on a measure space (X, X , μ) with some σ-finite non-atomic measure μ for which

∫ f_{j}^{2}(x_{(j,1)}, …, x_{(j,k_{j})})μ( dx_{(j,1)}) … μ( dx_{(j,k_{j})}) < ∞ for all 1 ≤ j ≤ m  (3.1)

together with a white noise μ_{W} with reference measure μ on the space (X, X). The Wiener–Itô integrals k_{j}!I_{k_{j}}(f_{j}), 1 ≤ j ≤ m, introduced in (1.3) can be defined with the above kernel functions f_{j} and white noise μ_{W}. We are interested in a good explicit

formula for the expectation E ( ∏_{j=1}^{m} I_{k_{j}}(f_{j})). This formula, which is a simple conse-

quence of the diagram formula for products of Wiener–Itô integrals (see [2] or [7]) will be presented below.

The expectation of the above product can be expressed by means of some (closed) diagrams introduced below. A class of (closed) diagrams denoted by Γ̄ = Γ̄(k_{1}, …, k_{m}) is defined in the following way. A diagram γ ∈ Γ̄(k_{1}, …, k_{m}) consists of vertices of the form (j, l), 1 ≤ j ≤ m, 1 ≤ l ≤ k_{j}, and edges ((j, l), (j′, l′)), 1 ≤ j, j′ ≤ m, 1 ≤ l ≤ k_{j}, 1 ≤ l′ ≤ k′_{j}. The set of vertices (j, l), 1 ≤ l ≤ k_{j}, with a fixed number j is called the j-th row of the diagram. All edges ((j, l), (j′, l′)) of a diagram γ ∈ Γ̄(k_{1}, …, k_{k}) connect vertices from different rows, i.e. j ≠ j′. It is also demanded that exactly one edge starts from all vertices of a (closed) diagram γ. The class Γ̄(k_{1}, …, k_{m}) of (closed) diagrams contains the diagrams γ with the above properties. Beside this, I introduce the set U(γ) containing the edges of the diagram γ for all γ ∈ Γ̄(k_{1}, …, k_{m}) and enumerate their elements. An arbitrary enumeration is allowed, it is only demanded that different edges must get different labels. Let N(γ) denote the set of indices attached to the elements of U(γ). In this section I shall consider only closed diagrams in which every vertex is the end-point of some edge. In the next section we have to work with more general, not necessarily closed diagrams. I introduce their definition there.

Let us fix a diagram γ ∈ Γ̄(k_{1}, …, k_{m}). I introduce the following function u_{γ}(·) on the vertices (j, l), 1 ≤ j ≤ m, 1 ≤ l ≤ k_{j}.

For each vertex of γ there is a unique edge ((j, l), (j′, l′)) ∈ U(γ), i.e. an edge in γ which connects (j, l) with some other vertex (j′, l′) of the diagram. If this edge has label n ∈ N(γ), then we put u_{γ}((j, l)) = n.

Given a fixed diagram γ ∈ Γ̄(k_{1}, …, k_{m}) let us rewrite the functions f_{j} with reindexed variables as f_{j}(x_{u_{γ}(j,1)}, …, x_{u_{γ}(j,k_{j})}), 1 ≤ j ≤ m, with the help of the above defined function u_{γ}(·). (Two variables get the same index if the vertices related to them were connected by an edge of the diagram γ.) Define the product of these reindexed variables

F̄_{γ}(x_{n}, n ∈ N(γ)) = ∏_{j=1}^{m} f_{j}(x_{u_{γ}(j,1)}, …, x_{u_{γ}(j,k_{j})})  (3.2)

together with the integral of these functions

F_{γ} = F_{γ}(f_{1}, …, f_{m}) = ∫ F̄_{γ}(x_{n}, n ∈ N(γ)) ∏_{n∈N(γ)} μ( dx_{n})  (3.3)

for all γ ∈ Γ̄(k_{1}, …, k_{m}).

The expected value of the product of Wiener–Itô integrals k_{j}!I_{k_{j}}(f_{j}), 1 ≤ j ≤ m, can be expressed with the help of the above quantities F_{γ} in the following way.

Formula about the expected value of products of Wiener–Itô integrals. Let us consider the Wiener–Itô integrals k_{j}!I_{k_{j}}(f_{j}) of some functions f_{j}, 1 ≤ j ≤ m, satisfying relation (3.1) with respect to a white noise μ_{W} with reference measure μ. The expected value of this product satisfies the identity

E (∏_{j=1}^{m} k_{j}!I_{k_{j}}(f_{j})) = ∑_{γ∈Γ̄(k_{1},...,k_{m})} F_{γ}  (3.4)

with the numbers F_{γ} defined in (3.2) and (3.3).

To get a good estimate on the expectation of the product of Wiener–Itô integrals by means of formula (3.4) we need a good bound on the quantities F_{γ}. For this goal it is useful to rewrite them by means of an appropriate recursive formula. To present such a formula let us define the restrictions γ^{r}, 1 ≤ r ≤ m, of all (closed) diagrams γ ∈ Γ̄(k_{1}, …, k_{m}) to its first r rows, 1 ≤ r ≤ m.

More explicitly, the diagram γ^{r} contains the vertices (j, l), 1 ≤ j ≤ r, 1 ≤ l ≤ k_{j} and those edges of γ whose end-points are vertices in one of the first r rows in γ. I shall call γ^{r} also a diagram, although it may have vertices from which no edge starts. I return to this point in the next section. Let U_{1}(γ^{r}) denote the set of edges and U_{2}(γ^{r}) the set of those vertices of γ^{r} from which no edge starts in γ^{r}, i.e. which are connected with a vertex (j′, l′) with j′ > r in γ. Let all vertices of γ^{r} get the same enumeration that they got as a vertex of γ. Let N_{1}(γ^{r}) denote the set of indices of the vertices which are end-points of an edge in U_{1}(γ^{r}), and N_{2}(γ^{r}) the set of indices of the vertices in U_{2}(γ^{r}). Put N(γ^{r}) = N_{1}(γ^{r}) ∪ N_{2}(γ^{r}).

Let us define, similarly to the quantities F̄_{γ} and F_{γ}, the functions

F̄_{γ^{r}}(x_{n}, n ∈ N(γ^{r})) = ∏_{j=1}^{r} f_{j}(x_{u_{γ}(j,1)}, …, x_{u_{γ}(j,k_{j})})  (3.5)

and

F_{γ^{r}}(x_{n}, n ∈ N_{2}(γ^{r})) = ∫ F̄_{γ^{r}}(x_{n}, n ∈ N(γ^{r})) ∏_{n∈N_{1}(γ_{r})} μ( dx_{n})  (3.6)

for all 1 ≤ r ≤ m, and γ ∈ Γ̄(k_{1}, …, k_{m}). In the case N_{1}(γ^{r}) = ∅ the integral at the right-hand side equals F̄_{γ^{r}}(x_{n}, n ∈ N(γ^{r})). In general, the following convention is applied. If we integrate a function with respect to a product measure, then in the case when this product contains zero terms, then the integral equals the function itself.

It is not difficult to check that

F_{γ^{1}}(x_{n}, n ∈ N_{2}(γ^{1})) = f_{1}(x_{u_{γ}(1,1)}, …, x_{u_{γ}(1,k_{1})}),  (3.7)

and

\begin{matrix}F_{γ^{r}}(x_{n}, n ∈ N_{2}(γ^{r})) \\ = ∫ F_{γ^{r−1}}(x_{n}, n ∈ N_{2}(γ^{r−1}))f_{r}(x_{u_{γ}(r,1)}, …, x_{u_{γ}(r,k_{r})}) ∏_{n∈N_{2}(γ^{r})\N_{2}(γ^{r−1})}μ( dx_{n}) \end{matrix}

(3.8)

for all 2 ≤ r ≤ m. Beside this,

F_{γ} = F_{γ^{m}}(x_{n}, n ∈ N_{2}(γ^{m})).  (3.9)

(Actually, N_{2}(γ^{m}) = ∅.) Relations (3.7), (3.8) and (3.9) yield a recursive formula for the quantity F_{γ}.

Next I introduce the notion of connected (closed) diagrams. They turned out to be a useful object, because the quantity F_{γ} can be better bounded for a connected than for a general diagram γ, and it is enough to bound them to prove our results.

Definition of connected (closed) diagrams. A (closed) diagram γ ∈ Γ̄(k_{1}, …, k_{m}) is connected if for all sets of rows A ⊂ {1, …, m} of the diagram γ such that 1 ≤ |A| ≤ m − 1 there is such an edge ((j_{1}, l_{1}), (j_{2}, l_{2})) ∈ U(γ) of the diagram γ for which j_{1} ∈ A and j_{2} ∉ A.

The following result, called the Basic Estimate yields a bound on F_{γ} for connected diagrams γ. This estimate turned out to be sufficient for our purposes.

Basic Estimate. Let us consider a connected, closed diagram γ ∈ Γ̄(k_{1}, …, k_{m}), m ≥ 2, and some functions f_{j} of k_{j} variables on a measure space (X, X , μ), 1 ≤ j ≤ m, which satisfy the inequality V_{s}(f_{j}) ≤ R^{s−1} with some 0 ≤ R ≤ 1, for all 1 ≤ j ≤ m and 1 ≤ s ≤ k_{j}. (The quantities V_{s}(f) were introduced in formulas (1.6) and (1.7).) The quantity F_{γ} = F_{γ}(f_{1}, …, f_{m}), introduced in (3.3) satisfies the inequality

|F_{γ}| = |F_{γ}(f_{1}, …, f_{m})| ≤ R^{m−2}.  (3.10)

I show how the simplified version of the theorem about the tail distribution of Wiener–Itô integrals can be proved with the help of the Basic Lemma. To do this first I show that all diagrams γ ∈ Γ̄(k_{1}, …, k_{m}) can be decomposed to the union of disjoint connected diagrams in a unique way, and the quantity F_{γ} equals the product of the numbers F_{{·}} corresponding to these connected diagrams.

More explicitly, there is a unique partition A_{1} = A_{1}(γ), …, A_{s} = A_{s}(γ) of the set of rows {1, …, m} of γ ∈ Γ̄(k_{1}, …, k_{m}) in such a way that the diagram γ equals the union of the diagrams γ_{A_{u}}, 1 ≤ u ≤ s, where γ_{A_{u}} is the restriction of γ to the rows in A_{u}, i.e. it contains the rows of γ with indices in the set A_{u} together with the edges connecting vertices from these rows. Beside this, all diagrams γ_{A_{u}} of this decomposition must be connected. Also the identity

F_{γ} = ∏_{u=1}^{s} F_{γ_{Au}}  (3.11)

holds.

In more detail, the restriction γ_{A_{u}} of γ to the rows with indices in A_{u} consists of vertices in the rows with index in the set of A_{u}, and those edges whose both end-points are among these vertices. The decomposition of γ to the diagrams γ_{A_{u}}, 1 ≤ u ≤ s, means also that γ has no such edge which connects vertices from γ_{A_{u}} with vertices from γ_{A_{u′}} with some u ≠ u′. The connectedness of γ_{A_{u}} means that for all subsets B ⊂ A_{u}, B ≠ ∅ and B ≠ A_{u} there is an edge of γ_{A_{u}} which connects a vertex in a row with index in B and a vertex in a row with index in A_{u} \ B. The quantities F_{γ_{Au}} can be defined similarly to F_{γ}, for instance by a natural adaptation of the recursive formulas (3.7), (3.8) and (3.9) in the calculation of F_{γ} to the calculation of F_{γ_{Au}}. In this adaptation we can write similar recursive relations, only the row indices 1, …, m must be replaced by rows with indices v_{1}, …, v_{|A_{u}|} if A_{u} = {v_{1}, …, v_{|A_{u}|}} with v_{1} < v_{2} < ⋯ < v_{|A_{u}|}. It is not difficult to check that the Basic Estimate also implies that under the conditions of this result the inequality |F_{γ_{A}}| ≤ R^{|A|−2} also holds for any connected (closed) diagram with rows in the set A ⊂ {1, …, m}.

To find the desired decomposition of a diagram γ ∈ Γ̄(k_{1}, …, k_{m}) let us define the graph G(γ) with vertices 1, …, m in which two vertices j_{1} and j_{2} are connected with an edge if and only if the diagram γ contains an edge which connects two vertices from the j_{1}-th and j_{2}-th rows. Let A_{1}, …, A_{s} be the connected disjoint components of this graph. Then it is not difficult to see that γ_{A_{1}}, …, γ_{A_{s}} supplies the desired decomposition of the diagram γ, and also relation (3.11) holds.

Given a set A ⊂ {1, …, m} let Γ̄_{c}(k_{j}, j ∈ A) denote the class of those connected (closed) diagrams whose rows are the sequences {(j, 1), …, (j, k_{j})}, j ∈ A. Put

K_{c}(A) = ∑_{γ_{A}∈Γ̄_{c}(k_{j}, j∈A)} F_{γ_{A}},  (3.12)

and let us also introduce for a partition P = {A_{1}, …, A_{s}} ∈ P({1, …, m}) of the set {1, …, m} the class of diagrams Γ̄(A_{1}, …, A_{s}) = Γ̄(A_{1}, …, A_{s}|k_{1}, …, k_{m}) containing those diagrams γ ∈ Γ̄(k_{1}, …, k_{m}) whose decomposition to connected components consists of diagrams with rows indexed by the sets A_{1}, …, and A_{s}, i.e. of diagrams γ_{A_{r}} ∈ Γ̄_{c}(k_{j}, j ∈ A_{r}), 1 ≤ r ≤ s. Define the quantity

K(A_{1}, …, A_{s}) = ∑_{γ∈Γ̄(A_{1},...,A_{s})} F_{γ}

for all partitions {A_{1}, …, A_{s}} of the set {1, …, m}.

It is not difficult to see with the help of relations (3.11) and (3.12) that

K(A_{1}, …, A_{s}) = K_{c}(A_{1}) ⋯ K_{c}(A_{s})

for all partitions {A_{1}, …, A_{s}} of the set {1, …, m}. Summing up this identity for all classes Γ̄(A_{1}, …, A_{r}) we get with the help of the identity (3.4) and the fact that each diagram has a unique decomposition to connected diagrams that

E (∏_{j=1}^{m} k_{j}!I_{k_{j}}(f_{j})) = ∑_{{A_{1},...,A_{s}}∈P({1,...,m})} K_{c}(A_{1}) ⋯ K_{c}(A_{s})  (3.13)

In the next calculations I shall restrict my attention to the case m = 2M, f_{j} = f, k_{j} = k for all 1 ≤ j ≤ 2M. I give a good upper bound on EI_{k}(f)^{2M} with the help of relations (3.12), (3.13) and the Basic Estimate in the case when V_{s}(f) ≤ R^{s−1} for all 1 ≤ s ≤ k with R = M^{−(k−1)/2}. In these calculations I also exploit that a connected closed diagram has at least two rows, hence only such partitions {A_{1}, …, A_{s}} ∈ P({1, …, m}) have to be considered in the sum at the right-hand side of (3.13) for which |A_{r}| ≥ 2 for all 1 ≤ r ≤ s.

First I show that if V_{s}(f) ≤ R^{s−1} for all 1 ≤ s ≤ k, then

|K_{c}(A)| ≤ (k|A|)^{k|A|/2}R^{|A|−2} for all A ⊂ {1, …, m}  (3.14)

for the quantity K_{c}(A) defined in (3.12). Indeed, the Basic Estimate implies that |F (γ_{A})| ≤ R^{|A|−2} for all terms in the sum at the right-hand side of (3.12), and clearly there are less than (k|A|)^{k|A|/2} diagrams with |A| rows and in each rows k vertices. (At this point we bounded the number of all (and not only the number of all connected) diagrams.)

To estimate the expression at the right-hand side of (3.13) let us introduce the class of those partitions P_{s,t_{1},...,t_{s}}, t_{1} + ⋯ + t_{s} = 2M, t_{r} ≥ 2, 1 ≤ r ≤ s, of the set {1, …, 2M} which consist of s sets A_{1}, …, A_{s}, and the set A_{r} has with t_{r} elements, 1 ≤ r ≤ s. (The set F_{s,t_{1},...,t_{s}} depends on the number s and the set {t_{1}, …, t_{s}}, but it does not depend on the order of the elements t_{r} in this set.) Let us first estimate the contribution of the partitions P_{s,t_{1},...,t_{s}} to the sum in (3.13). I claim that

\begin{matrix}|  ∑_{\{A_{1},...,A_{s}\}∈P_{s,t_{1},...,ts}} K_{c}(A_{1}) ⋯ K_{c}(A_{s})| \\ ≤ |P_{s,t_{1},...,t_{s}}| ∏_{r=1}^{s}(kt_{r})^{kt_{r}/2}R^{t_{r}−2} ≤ C^{M}M^{M} [M^{(k−1)}R^{2}]^{M−s} \end{matrix}  (3.15)

with a universal constant C > 0 depending only on the parameter k. In particular,

|  ∑_{{A_{1},...,A_{s}}∈P_{s,t_{1},...,ts}} K_{c}(A_{1}) ⋯ K_{c}(A_{s})| ≤ C^{M}M^{M}  (3.16)

if V_{s}(f) ≤ M^{−(k−1)(s−1)/2} for all 1 ≤ s ≤ k, because (3.15) holds with R = M^{−(k−1)/2} in this case.

The first inequality in (3.15) is a straight consequence of relation (3.14). To prove the second inequality we need a good bound on |P_{s,t_{1},...,t_{s}}|. To get such a bound let us first list the element A_{1}, …, A_{s} of a partition in P_{s,t_{1},...,t_{s}} in the following way. Let A_{1} be the set which contains the number 1, A_{2} the set containing the smallest number not contained in A_{1}, e.t.c.. Then let t_{r} denote the cardinality of the set A_{r} with such an indexation. Then the number |P_{s,t_{1},...,t_{s}}| can be given explicitly, but the following estimate will be sufficient for our purposes.

|P_{s,t_{1},...,t_{s}}| ≤ ∏_{r=1}^{s} \frac{(2M)^{t_{r}−1}}{(t_{r} − 1)!} ≤ C_{1}^{M} \frac{M^{2M−s}}{∏_{r=1}^{s} t_{r}^{t_{r}}}  (3.17)

with some appropriate C_{1} > 0.

The first inequality in relation (3.17) can be simply checked. To prove the second

inequality let us first observe that ∏_{r=1}^{s} t_{r} ≤ (\frac{1}{s} ∑_{r=1}^{s} t_{r})^{s} = (\frac{2M}{s} )^{s} ≤ C_{2}^{M} with some

universal constant C_{2}. This inequality together with the Stirling formula imply that

∏_{r=1}^{s}(t_{r} − 1)! ≥ C_{2}^{−M} ∏_{r=1}^{s} t_{r}! ≥ C_{3}^{−M} ∏_{r=1}^{s} t_{r}^{t^{r}} with some appropriate values C_{3}, hence

relation (3.17) holds.

It follows from relation (3.17) that

|P_{s,t_{1},...,t_{s}}| ∏_{r=1}^{s}(kt_{r})^{kt_{r}/2}R^{t_{r}−2} ≤ C_{4}^{M}M^{2M−s} (∏_{r=1}^{s} t_{r}^{(k−2)t_{r}/2}) R^{2M−2s}.  (3.18)

Let us consider the maximum of the right-hand side in the inequality (3.18) in the parameters t_{1}, …, t_{s} with a fixed value s. Since k − 2 ≥ 0, this expression takes its maximum if t_{1} = 2M − 2s + 2 and t_{r} = 2, 2 ≤ r ≤ s. Hence

|P_{s,t_{1},...,t_{s}}| ∏_{r=1}^{s}(kt_{r})^{kt_{r}/2}R^{t_{r}−2}

≤ C_{4}^{M}M^{2M−s}(2M − 2s + 2)^{(k−2)(M−s+1)}2^{(s−1)(k−1)}R^{2M−2s}
≤ C^{M}M^{2M−s}M^{(k−2)(M−s)}R^{2M−2s} = C^{M}M^{M} [M^{k−1}R^{2}]^{M−s}

with some C > 0 depending only on the parameter k. This inequality finishes the proof of relation (3.15).

Relation (3.16) is a simple consequence of (3.15). Finally, it follows from (3.4) (with m = 2M and f_{j} = f for all 1 ≤ j ≤ 2M) and (3.16) that

E(k!I_{k}(f))^{2M} ≤ 2^{2M}C^{M}M^{M}

if V_{s}(f) ≤ M^{−(k−1)(s−1)/2} for all 1 ≤ s ≤ k, which means that the simplified version of the theorem about the tail distribution of Wiener–Itô integrals is proved with the help of the Basic Estimate.

To see that relations (3.4) and (3.16) really imply the last inequality it is enough to observe that the class of sets P({1, …, 2M}) is the union of the classes of sets P_{s,t_{1},...,t_{s}}, and there are less than 2^{2M} classes P_{s,t_{1},...,t_{s}}. (The number of such classes is bounded by the number of sets of positive integers {t_{1}, …, t_{s}} such that t_{1} + ⋯ + t_{s} = 2M.)

We could prove our result by means of a good estimate on the quantity F_{γ} for connected diagrams. The introduction of connected diagrams was very important in our considerations, because the Basic Estimate holds only for such diagrams. I show an example of non-connected (closed) diagrams which satisfies only a much weaker estimate. Let us consider a diagram γ ∈ Γ(k_{1}, …, k_{m}) and a function f_{j} = f with m = 2M, k_{j} = k for all 1 ≤ j ≤ 2M. Beside this, let the f be a function, symmetric in its variables and such that V_{1}(f) = 1. Let us consider such a diagram γ whose rows can be put into pairs in such a way that edges can connect only vertices from rows which are paired. For such a diagram F_{γ} = V_{1}(f)^{2M} = 1, and this value is much larger than the bound in the Basic Estimate. But there are relatively few such diagrams, their number equals \frac{(2M)!}{2^{M} M!}(k!)^{M}. Hence the relatively great value of F_{γ} for such diagrams causes no problem.

The estimation of the moments with the help of connected diagrams corresponds to a classical method of probability theory, to the estimation of moments by means of semi-invariants. The quantity K_{c}(A) introduced in formula (3.12) is actually the semiinvariant of the random variables k_{j}!I_{k_{j}}(f) for j ∈ A, and the identity (3.13) is a special case of the formula about the expression of the expectation of products of random variables by means of semi-invariants. Since I shall not work with semi-invariants in an explicit form in this paper, here I do not discuss this subject in more detail. Some useful information about semi-invariants can be found in the second chapter of the book [11], and I also return to this question in paper [10].

4. The proof of the Basic Estimate.

The Basic Estimate will be proved by means of an inductive proposition about the behaviour of the functions F_{γ^{r}}, 1 ≤ r ≤ m, where γ^{r} is the restriction of the closed diagram γ to its first r rows. To formulate and prove this result the notion of diagrams, introduced in Section 3, will be generalized. Such diagrams will be defined which may have such vertices from which no edge starts. The definition of some other objects related to these new diagrams will be also presented together with some results needed in our further discussion.

A diagram with rows in a finite set A = {j_{1}, …, j_{m}}, 1 ≤ j_{1} < j_{2} < ⋯ < j_{m} of the positive integers and with row length j_{t}, 1 ≤ t ≤ m, is a graph whose vertices are the pairs (j_{t}, l) with 1 ≤ t ≤ m and 1 ≤ l ≤ k_{j_{t}}. The set of points {(j_{t}, l): 1 ≤ l ≤ k_{j_{t}}} is called the t-th row of the diagram. A diagram may contain such edges ((j_{t}, l), (j_{t′}, l′)) for which j_{t}, j_{t′} ∈ A, 1 ≤ l ≤ k_{j_{t}}, 1 ≤ l′ ≤ k_{j_{t′}}, and j_{t} ≠ j_{t′}. In words this means that edges can connect only vertices from different rows. Beside this, it is required that from each vertex there starts either zero or one edge. Graphs satisfying all these properties will be called diagrams. The class of diagrams with rows indexed by a set A and with k_{j} element in the row indexed by j ∈ A will be denoted by Γ(k_{j}|j ∈ A). The main difference between diagrams and closed diagrams introduced in the previous section is that a general diagram may contain vertices from which no edge starts. Those vertices of a diagram γ from which no edge starts will be called open vertices. The notion of connected diagrams will be also introduced for this more general class of diagrams.

Definition of connected diagrams in the general case. A diagram γ ∈ Γ(k_{j}|j ∈ A) is connected if for all sets B ⊂ A such that B ≠ ∅ and B ≠ A there exists an edge ((j_{t}, l), (j_{t′}, l′)) of γ such that j_{t} ∈ B and j_{t′} ∈ A \ B.

Similarly to closed diagrams, general diagrams γ ∈ Γ(k_{j}|j ∈ A) also have a unique decomposition to connected diagrams. To formulate this statement precisely let us first introduce the reduction of a diagram to some of its rows. Given a diagram γ ∈ Γ(k_{j}|j ∈ A), A = {j_{1}, …, j_{m}} and a set B ⊂ A let γ_{B} ∈ Γ(k_{j}|j ∈ B) denote that diagram whose vertices are points of the form (j_{t}, l), j_{t} ∈ B, 1 ≤ l ≤ k_{j_{t}}, and two vertices in γ_{B} are connected by an edge if and only if they are connected by an edge in γ. With this notation we can state that for all diagrams γ ∈ Γ(k_{j}|j ∈ A) there exists a unique partition A_{1}, …, A_{s} of the set of rows A in such a way that all diagrams γ_{A_{u}}, 1 ≤ u ≤ s, are connected, and the diagram γ_{A} is the union of the diagrams γ_{A_{u}}, 1 ≤ u ≤ s. This can be proved similarly to the case of closed diagrams. One has to take the graph whose vertices are the points of the set A and draw an edge between two vertices j_{t} ∈ A, and j_{t′} ∈ A if there is an edge in the diagram γ connecting some vertices from the j_{t}-th and the j_{t′}-th row. By taking the decomposition of this graph to connected components we also get the decomposition of the diagram γ to connected components.

Given a diagram γ ∈ Γ(k_{j}|j ∈ A) with A = {j_{1}, …, j_{m}} together with some functions f_{t}(x_{(j_{t},1)}, …, x_{(j_{t},k_{j_{t}})}), 1 ≤ t ≤ m, a natural generalization of the quantities F̄_{γ} and F_{γ} defined in (3.2) and (3.3) can be introduced, and it can be shown that they have properties similar to them. To introduce these quantities let us enumerate first the edges and then the vertices of the diagram γ. A vertex from which an edge starts gets the same label as the edge starting from it. The remaining vertices, from which no edge starts get a new label. In this enumeration two vertices get the same index if and only if they are connected by an edge. Let u_{γ}(j, l) denote the label of of the vertex (j, l) in this enumeration of the vertices of a diagram γ. Let N(γ) denote the set of labels of all vertices and N_{1}(γ) the set of labels of all open vertices in γ. Then we define, similarly to formulas (3.2) and (3.3) the quantities

F̄_{γ}(x_{n}, n ∈ N(γ)) = ∏_{t=1}^{m} f_{t}(x_{u_{γ}(j_{t},1)}, …, x_{u_{γ}(j_{t},k_{j_{t}})})  (4.1)

and

\begin{aligned}F_{γ}(x_{n}, n ∈ N_{1}(γ)) & = F_{γ}(x_{n}, n ∈ N_{1}(γ)|f_{1}, …, f_{m}) \\ & = ∫ F̄_{γ}(x_{n}, n ∈ N(γ)) ∏_{n∈N_{2}(γ)} μ( dx_{n}) \end{aligned}  (4.2)

with N_{2}(γ) = N(γ) \ N_{1}(γ).

Given a diagram γ ∈ Γ(k_{j}|j ∈ A) with A = {j_{1}, …, j_{m}} let γ^{r} denote its restriction to its first r rows, i.e. to rows in A_{r} = {j_{1}, …, j_{r}}, 1 ≤ r ≤ m. The natural modification of the recursive relations (3.7), (3.8) and (3.9) remain valid also for general diagrams γ. The only difference we have to make is to rewrite the indices u_{γ}(1, l) and u_{γ}(r, l) by u_{γ}(j_{1}, l) and u_{γ}(j_{r}, l) in the new relations. Beside this, the following natural modification of formula (3.11) holds.

If the decomposition of a diagram γ = γ_{A} ∈ Γ(k_{j}|j ∈ A) to connected components consists of the diagrams γ_{A_{1}}, …, γ_{A_{s}}, where A_{1}, …, A_{s} is the partition of the set A we have to apply to get the desired decomposition to connected components, then the above defined function F_{γ}(x_{n}, n ∈ N_{1}(γ)) satisfies the identity

F_{γ_{A}}(x_{n}, n ∈ N_{1}(γ_{A})) = ∏_{t=1}^{s} F_{γ_{A_{t}}} (x_{n}, n ∈ N_{1}(γ_{A_{t}})),  (4.3)

where N_{1}(γ_{A_{t}}) denotes the set of labels of the open vertices in γ_{A_{t}}, and F_{γ_{A_{t}}} is the function defined in (4.2) with the diagram γ = γ_{A_{t}}. Beside this, the sets of labels N_{1}(γ_{A_{t}}) are disjoint for different indices t. Here I make the convention that the label of a vertex in the restriction γ_{B} of a diagram γ_{A}, B ⊂ A, agrees with its original label in the diagram γ_{A}. Formula (4.3) can be checked similarly to (3.8) with the help of (the modified version of) the recursive relations (3.8) and (3.9).

Given a closed diagram γ, its restriction γ^{r} to its first r rows may be not a closed diagram. But it is also a diagram, and the above mentioned results can be applied for it. In particular, it has a decomposition to connected components, and the function F_{γ_{r}} defined by means of some functions f_{j}, 1 ≤ j ≤ r can be factorized. The next result, called A More Detailed Version of the Basic Estimate contains an estimate for the terms in this factorization. The Basic Estimate is a part of this result for the parameter r = m.

A More Detailed Version of the Basic Estimate. Let us consider a connected, closed diagram γ ∈ Γ̄(k_{1}, …, k_{m}), m ≥ 2, and some functions f_{j} of k_{j} variables on a measure space (X, X , μ), 1 ≤ j ≤ m, which satisfy the inequality V_{s}(f_{j}) ≤ R^{s−1} with some 0 ≤ R ≤ 1 for all 1 ≤ j ≤ m and 1 ≤ s ≤ k_{j}. Let γ^{r} denote the restriction of the diagram γ to its first r rows and let F_{γ^{r}} be the function defined in formula (4.2) with the help of the diagram γ = γ^{r} and the functions f_{s}, 1 ≤ s ≤ r. Take the decomposition of the diagram γ^{r} to the union of disjoint connected diagrams γ_{A_{t}^{r}} defined with the help of an appropriate partition A_{1}^{r}, …, A_{s(r)}^{r} of the set {1, …, r}, 1 ≤ t ≤ s(r), with some number s(r). Consider the factorization (4.3) of the function F_{γ^{r}} to

F_{γ^{r}}(x_{n}, n ∈ N_{1}(γ^{r})) = ∏_{t=1}^{s(r)} F_{γ_{A^{r}_{t}}} (x_{n}, n ∈ N_{1}(γ_{A_{t}^{r}})).

For 1 ≤ r ≤ m − 1 all diagrams γ_{A_{t}^{r}}, 1 ≤ t ≤ s(r), contain at least one open vertex, i.e. N_{1}(γ_{A_{t}^{r}}) ≥ 1, and

|V_{s}(F_{γ_{A^{r}_{t}}} (x_{n}, n ∈ N_{1}(γ_{A_{t}^{r}})))| ≤ R^{|A_{t}^{r}|+s−2} for all 1 ≤ t ≤ s(r) and 1 ≤ s ≤ |N_{1}(γ_{A_{t}^{r}})|,

(4.4)

where |A_{t}^{r}| is the cardinality of the set A_{t}^{r}, i.e. it equals the number of rows in the diagram A_{t}^{r}, and N_{1}(γ_{A_{t}^{r}}) denotes the set of indices of the free vertices in γ_{A_{t}^{r}}.

For r = m the decomposition of γ^{m} = γ to connected components consists of the (closed) diagram γ itself, and

|F_{γ}| = |F_{γ^{m}}| ≤ R^{m−2}.  (4.5)

The proof of the More Detailed Version of the Basic Estimate is based on the following result called the Main Inequality.

The Main Inequality. Let f(x_{1}, …, x_{n}, v_{n+m+1}, …, v_{n+m+p}) and

g(x_{n+1}, …, x_{n+m}, v_{n+m+1}, …, v_{n+m+p})

be two square integrable functions with m + p and n + p variables on a measure space (X, X , μ), and define the function

\begin{aligned}F (x_{1}, …, x_{n+m}) & = ∫ f(x_{1}, …, x_{n}, v_{n+m+1}, …, v_{n+m+p}) \\ & g(x_{n+1}, …, x_{n+m}, v_{n+m+1}, …, v_{n+m+p}) ∏_{u=n+m+1}^{n+m+p} μ( dv_{u}). \end{aligned} (4.6)

Let m+n ≥ 1, p ≥ 1, and let the functions f and g satisfy the relation V_{s}(f) ≤ D_{1}R^{s−2} and V_{s}(g) ≤ D_{2}R^{s−2} with some A > 0, B > 0 and 0 ≤ R ≤ 1 for all 1 ≤ s ≤ m + p and 1 ≤ s ≤ n + p respectively. Then the function F satisfies the inequality

V_{s}(F (x_{1}, …, x_{n+m})) ≤ D_{1}D_{2}R^{s−2} for all 1 ≤ s ≤ m + n.  (4.7)

First I prove the More Detailed Version of the Basic Estimate with the help of the Main Inequality.

Proof of the More Detailed Version of the Basic Estimate with the help of the Main Inequality. Let us first observe that all components γ_{A_{t}^{r}}, 1 ≤ t ≤ s(r), of the diagram γ^{r} contain at least one open vertex for r ≤ m − 1. Otherwise the diagram γ would not satisfy the condition of connectedness with B = A_{t}^{r}. Relation (4.4) clearly holds for r = 1. It will be proved by induction with respect to r that it holds for all 1 ≤ r ≤ m−1.

Let us assume that relation (4.4) holds for r − 1 and let us prove it for r. Some of the connected components γ_{A_{t}^{r−1}} of γ^{r−1} may have a vertex connected with a vertex of the r-th row {(r, 1), …, (r, k_{r})} of the diagram γ. Let us make such an enumeration of the connected components of γ^{r−1} in which there is a constant s̄(r) such that the components γ_{A_{t}^{r−1}} with 1 ≤ s̄(r) have a vertex connected with a vertex of the r-th row of γ, and the components γ_{A_{t}^{r−1}}, s̄(r) < t ≤ s(r − 1), have no such vertices. To simplify the following discussion let us introduce the notation γ_{A_{0}^{r−1}} = {(r, 1), …, (r, k_{r})} for the r-th row of the diagram γ. Then the connected components of γ^{r} are the diagrams

γ_{B^{r}} = ⋃_{t=0}^{s̄(r)} γ_{A_{t}^{r−1}} and γ_{A_{t}^{r−1}}, s̄(r) < t ≤ s(r − 1). The latter diagrams satisfy relation

(4.4) by induction, so it is enough to show that

|V_{s}(F_{γ_{B^{r}}})| ≤ R^{|B^{r}|+s−2} for all 1 ≤ s ≤ |N_{1}(γ_{B^{r}})|,  (4.8)

where |N_{1}(γ_{B^{r}})| denotes the number of open vertices in the diagram γ_{B^{r}}. It is possible that no vertex of the diagram γ^{r−1} is connected with a vertex from the r-th row of γ. In this case s̄(r) = 0, B^{r} = A_{0}^{r−1}, and relation (4.8) clearly holds.

To prove relation (4.8) let us introduce the following notations. Put B_{j}^{r} = ⋃_{t=0}^{j} A_{t}^{r−1}

for all 0 ≤ j ≤ s̄(r). With such a notation B_{s̄(r)}^{r} = B^{r} for j = s̄(r). I shall show that

|V_{s}(F_{γ_{B^{r}_{j}}} (x_{s}, s ∈ N_{1}(γ_{B_{j}^{r}})| ≤ R^{|B_{j}^{r}|+s−2} for all 1 ≤ j ≤ s̄(r) and 1 ≤ s ≤ |N_{1}(γ_{B_{j}^{r}})|.

(4.9)

Relation (4.9) for j = s̄(r) implies relation (4.8). Relation (4.9) holds for j = 0, and it will be proved for a general parameter j, 1 ≤ j ≤ s̄(r), by induction.

For this goal I write the following recursive relation for the functions F_{γ_{B^{r}_{j}}}

\begin{aligned}F_{γ_{B^{r}_{j}}} (x_{s}, s ∈ N_{1}(γ_{B_{j}^{r}})) & = ∫ F_{γ_{A{r−1 \atop j}}} (x_{s}, s ∈ N_{1}(γ_{A_{j}^{r−1}})) \\ & F_{γ_{B{r \atop j−1}}} (x_{s}, s ∈ N_{1}(γ_{B_{j−1}^{r}})) ∏_{s∈N_{1}(γ_{A{r−1 \atop j}})∩N_{1}(γ_{B{r \atop j−1}} )} μ( dx_{s}) \end{aligned}

(4.10)

for all 1 ≤ j ≤ s̄(r). I show that relation (4.9) follows from relation (4.10) and the Main Inequality. Indeed, let us apply the Main Inequality with the functions f = F_{γ_{A{r−1 \atop j}}} (x_{s}, s ∈ N_{1}(γ_{A_{j}^{r−1}})) and g = F_{γ_{B{r \atop j−1}}} (x_{s}, s ∈ N_{1}(γ_{B_{j−1}^{r}})). (More precisely, we

apply an equivalent version of the Main Inequality where the indices of the variables of the functions f and g may be different, and the variables by which we integrate and by which we do not integrate may be listed in an arbitrary order.) By our inductive hypothesis these functions f and g satisfy the inequalities V_{s}(f) ≤ D_{1}R^{s−2} and V_{s}(g) ≤ D_{2}R^{s−2} with D_{1} = R^{|A_{j}^{r−1}|} and D_{2} = R^{|B_{j−1}^{r}|}. Beside this, |N_{1}(γ_{A_{j}^{r−1}}) ∩ N_{1}(γ_{B_{j−1}^{r}})| = |N_{1}(γ_{A_{j}^{r−1}})∩N_{1}(γ_{A_{0}^{r−1}})| ≥ 1, because there is an edge connecting a vertex of γ_{A_{j}^{r−1}} with a vertex of the r-th row γ_{A_{0}^{r−1}} of the diagram γ. This inequality corresponds to the condition p ≥ 1 in the Main Inequality. This number p is the multiplicity of the integral in formula (4.6). Beside this, the diagram γ_{A_{j}^{r−1}} ∪ γ_{B_{j−1}^{r}} has an open vertex because of the connectedness of the diagram γ. This corresponds to the condition m + n ≥ 1 in the Main Inequality.

The above considerations show that the Main Inequality can be applied in the present case. It yields that |V_{s}(F_{γ_{B^{r}_{j}}} | ≤ R^{|A_{j}^{r−1}|+|B_{j−1}^{r}|}R^{s−2} = R^{|B_{j}^{r}|+s−2}, and this is what we had to prove.

The proof of relation (4.5) is similar. In the proof the above decomposition of γ^{r−1} is applied to the connected components for r = m. For this parameter r = m all components γ_{A_{t}^{m−1}}, 1 ≤ t ≤ s(m − 1) have a vertex which is connected with a vertex of the m-th row of the diagram γ. (The m-th row of γ will be sometimes denoted by γ_{0}^{m−1}.) Thus s̄(m) = s(m − 1), the connected components γ_{A_{j}^{m−1}}, 1 ≤ j ≤ s(m − 1), can be listed in an arbitrary order, and the diagrams γ_{B_{j}^{m}}, 1 ≤ j ≤ s(m − 1) can be defined similarly to the case r < m. Beside this, relation (4.9) can be proved for r = m and j ≤ s(m−1)−1, similarly as it was proved for r < m. The main difference between the cases r < m and r = m is that in the latter case the proof of relation (4.9) works only for j ≤ s(m − 1) − 1, but not for j = s(m − 1). In the case j = s(m) the Main Inequality cannot be applied in the proof, because γ_{B_{s(m)}^{m}} = γ is a diagram without open vertices, and this property does not allow the application of the Main Inequality in the case r = m, j = s(m − 1).

On the other hand the identity

F_{γ} = ∫ F_{γ_{B{m \atop s(m−1)−1}}} (x_{s}, s ∈ N_{1}(γ_{A_{s(m−1)}^{m−1}}))

F_{γ_{A{m−1 \atop s(m−1)}}} (x_{s}, s ∈ N_{1}(γ_{A_{s(m−1)}^{m−1}})) ∏_{s∈N_{1}(γ_{A{m−1 \atop s(m−1)}} )} μ( dx_{s})  (4.11)

holds, and the function integrated in (4.11) is the product of two terms whose L_{2}-norm can be well bounded. Namely, since the L_{2}-norm of a function f of several variables (defined on the measure space (X, X , μ)) equals V_{1}(f), relation (4.9) with the choice r = m and j = s(m − 1) − 1 together with formula (4.4) for r = m − 1 yield, with the parameter s = 1, the bound R^{|B_{s(m−1)−1}^{m}|−1} and R^{|A_{s(m−1)}^{m−1}|−1} for the L_{2}-norm of these terms. Hence relation (4.11) and the Schwarz inequality imply that

|F_{γ}| ≤ R^{|B_{s(m−1)−1}^{m}|+A_{s(m−1)}^{m−1}|)−2} = R^{m−2},

as it was stated.

It remained to prove the Main Inequality.

Proof of the Main Inequality. Let us fix a partition A_{j} = A_{j}^{(1)} ∪ A_{j}^{(2)}, 1 ≤ j ≤ s_{1}, B_{k}, 1 ≤ k ≤ s_{2}, and C_{l}, 1 ≤ l ≤ s_{3}, s_{1} + s_{2} + s_{3} = s of the set {1, …, n + m} such that all sets A_{j}^{(1)}, A_{j}^{(2)}, B_{k} and C_{l} are non-empty, and A_{j}^{(1)}, B_{k} ⊂ {1, …, n}, A_{j}^{(2)}, C_{l} ⊂ {n + 1, …, n + m}. Define some functions u_{j}(x_{q}, q ∈ A_{j}), 1 ≤ j ≤ s_{1}, v_{k}(x_{r}, r ∈ B_{k}),

1 ≤ k ≤ s_{2} and w_{l}(x_{t}, t ∈ C_{l}), 1 ≤ l ≤ s_{3} such that ∫ u_{j}^{2}(x_{q}, q ∈ A_{j}) ∏_{q∈A_{j}} μ( dx_{q}) ≤ 1,

∫ v_{k}^{2}(x_{r}, r ∈ B_{k}) ∏_{r∈B_{k}} μ( dx_{r}) ≤ 1, and ∫ w_{l}^{2}(x_{t}, t ∈ C_{l}) ∏_{t∈C_{l}} μ( dx_{t}) ≤ 1. It has to be

shown that for all such partitions and the functions f and g the inequality

\begin{matrix}∫ f(x_{1}, …, x_{n}, v_{n+m+1}, …, v_{n+m+p})g(x_{n+1}, …, x_{n+m}, v_{n+m+1}, …, v_{n+m+p}) \\ ∏_{j=1}^{s_{1}} u_{j}(x_{q}, q ∈ A_{j}) ∏_{k=1}^{s_{2}} v_{k}(x_{r}, r ∈ B_{k}) ∏_{l=1}^{s_{3}} w_{l}(x_{t}, t ∈ C_{l}) \\ ∏_{i=1}^{n+m} μ( dx_{i}) ∏_{u=n+m+1}^{n+m+p} μ( dv_{u}) ≤ D_{1}D_{2}R^{s−2} \end{matrix} (4.12)

holds.

Let us first show that this inequality can be reduced to the case when s_{2} = s_{3} = 0, that is when the partition of the set {1, …, n + m} consists only of such sets A_{j} which have a non-empty intersection with both sets {1, …, n} and {n + 1, …, n + m}. To carry out this reduction let us define the functions

\begin{matrix}f̄(x_{j}, j ∈ \{1, …, n\} \ B, v_{n+m+1}, …, v_{n+m+p}) \\ = ∫ f(x_{1}, …, x_{n}, v_{n+m+1}, …, v_{n+m+p}) ∏_{k=1}^{s_{2}} v_{k}(x_{r}, r ∈ B_{k}) ∏_{r∈B} μ( dx_{r}), \end{matrix} (4.13)

\begin{matrix}ḡ(x_{j}, j ∈ \{n + 1, …, n + m\} \ C, v_{n+m+1}, …, v_{n+m+p}) \\ = ∫ g(x_{n+1}, …, x_{n+m}, v_{n+m+1}, …, v_{n+m+p}) ∏_{l=1}^{s_{3}} w_{l}(x_{t}, t ∈ C_{l}) ∏_{t∈C} μ( dx_{t}) \end{matrix}

(4.14)

with B = ⋃_{k=1}^{s_{2}} B_{k} and C = ⋃_{l=1}^{s_{2}} C_{l} together with the function

F̄(x_{j}, j ∈ {1, …, n + m} \ (B ∪ C)

= ∫ f̄(x_{j}, j ∈ {1, …, n} \ B, v_{n+m+1}, …, v_{n+m+p})

ḡ(x_{j}, j ∈ {n + 1, …, n + m} \ C, v_{n+m+1}, …, v_{n+m+p}) ∏_{u=n+m+1}^{n+m+p} μ( dv_{u}).

(4.15)

With this notation inequality (4.12) can be rewritten as

\begin{aligned}∫ & F̄(x_{i}, i ∈ \{1, …, n + m\} \ (B ∪ C) ∏_{j=1}^{s_{1}} u_{j}(x_{q}, q ∈ A_{j}) \\ & ∏_{i∈\{1,...,n+m\}\(B∪C)} μ( dx_{i}) ≤ D_{1}D_{2}R^{s_{1}+s_{2}+s_{3}−2}. \end{aligned}  (4.16)

Beside this, it is not difficult to check that the functions f̄ and ḡ satisfy the inequalities V_{s}(f̄) ≤ D̄_{1}R^{s−2} for all 1 ≤ s ≤ n+p−|B| and V_{s}(ḡ) ≤ D̄_{2}R^{s−2} for all 1 ≤ s ≤ m+p−|C| with D̄_{1} = D_{1}R^{s_{2}} and D̄_{2} = D_{2}R^{s_{3}} respectively. For the parameter s = 1 these inequalities yield the bounds D̄_{1}R^{−1} and D̄_{2}R^{−1} fpr the L_{2}-norm of the functions f̄ and ḡ respectively. They imply together with relation (4.15) and the Schwarz inequality that relation (4.16) holds in the special case s_{1} = 0, i.e. when there is no set of the type A_{j} in the partition we consider. (In this case F̄ is a function of zero variables, i.e. it is a constant.)

This means that it is enough to prove relation (4.16) in the case s_{1} ≥ 1. This enables us to reduce the proof of relation (4.12) to the case s_{2} = s_{3} = 0, i.e. to the case when all elements of the partition is such a set as the sets A_{j}. To get this reduction it is enough to work with the functions f̄, ḡ and F̄ defined in formulas (4.13), (4.14) and (4.15) instead of the original functions f, g and F in the proof of the Main Inequality, and observe that these functions also satisfy its conditions (with an appropriate reindexation of the variables in these functions). Hence the Main inequality follows from the following reduced form of relation (4.12):

\begin{aligned}I & = ∫ f(x_{1}, …, x_{n}, v_{n+m+1}, …, v_{n+m+p})g(x_{n+1}, …, x_{n+m}, v_{n+m+1}, …, v_{n+m+p}) \\ & ∏_{j=1}^{s} u_{j}(x_{q}, q ∈ A_{j}) ∏_{i=1}^{n+m} μ( dx_{i}) ∏_{u=n+m+1}^{n+m+p} μ( dv_{u}) ≤ D_{1}D_{2}R^{s−2} \end{aligned}

(4.17) for a partition A_{j} = A_{j}^{(1)} ∪ A_{2}^{(j)}, 1 ≤ j ≤ s, of the set {1, …, n + m}. To prove this inequality let us introduce the functions

U_{j}(x_{q}, q ∈ A_{j}^{(2)}) = [∫ u_{j}^{2}(x_{q}, q ∈ A_{j}) ∏_{r∈A_{j}^{(1)}} μ( dx_{r})]^{1/2}

and

ū_{j}(x_{q}, q ∈ A_{j}^{(1)}|x_{q}, q ∈ A_{j}^{(2)}) = \frac{u_{j}(x_{q}, q ∈ A_{j})}{U_{j}(x_{q}, q ∈ A_{j}^{(2)})}

for all 1 ≤ j ≤ s together with the functions

G(x_{n+1}, …, x_{n+m})

= [∫ g^{2}(x_{n+1}, …, x_{n+m}, v_{n+m+1}, …, v_{n+m+p}) ∏_{u=n+m+1}^{n+m+p} μ( dv_{u})]^{1/2}

and

ḡ(v_{n+m+1}, …, v_{n+m+p})|x_{n+1}, …, x_{n+m}) = \frac{g(x_{n+1}, …, x_{n+m}, v_{n+m+1}, …, v_{n+m+p})}{G(x_{n+1}, …, x_{n+m})}.

Observe that

∫ U_{j}^{2}(x_{q}, q ∈ A_{j}^{(2)}) ∏_{q∈A_{j}^{(2)}} μ( dx_{q}) ≤ 1,  (4.18)

∫ ū_{j}^{2}(x_{q}, q ∈ A_{j}^{(1)}|x_{q}, q ∈ A_{j}^{(2)}) ∏_{q∈A_{j}^{(1)}} μ( dx_{q}) = 1 for all x = {x_{q}, q ∈ A_{j}^{(2)}} (4.19)

for all 1 ≤ j ≤ s, because the L_{2}-norm of the functions u_{j}(·) are less than 1. Similarly,

∫ G^{2}(x_{n+1}, …, x_{n+m}) ∏_{j=n+1}^{n+m} μ( dx_{j}) ≤ D_{2}^{2}R^{−2},  (4.20)

since the L_{2}-norm of the function G equals the L_{2}-norm of g which is U_{1}(g), and it is bounded by D_{2}R^{−1}. Beside this,

\begin{aligned}∫ & ḡ^{2}(v_{n+m+1}, …, v_{n+m+p}|x_{n+1}, …, x_{n+m}) ∏_{u=n+m+1}^{n+m+p} μ( dv_{u}) = 1 \\ & for all x = (x_{n+1}, …, x_{n+m}) \end{aligned}  (4.21)

The function

Z(x_{r}, n + 1 ≤ r ≤ n + m) = ∫ f(x_{1}, …, x_{n}, v_{n+m+1}, …, v_{n+m+p})

∏_{j=1}^{s} ū_{j}(x_{q}, q ∈ A_{j}^{(1)}|x_{q}, q ∈ A_{j}^{(2)})

ḡ(v_{n+m+1}, …, v_{n+m+p}|x_{n+1}, …, x_{n+m}) ∏_{i=1}^{n} μ( dx_{i}) ∏_{u=n+m+1}^{n+m+p} μ( dv_{u})

satisfies the inequality

|Z(x_{r}, n + 1 ≤ r ≤ n + m)| ≤ D_{1}R^{s−1} for all x = (x_{r}, n + 1 ≤ r ≤ n + m), (4.22)

because of the inequality V_{s+1}(f) ≤ D_{1}R^{s−1} (we consider the partition of the index set consisting of the sets A^{(1)}, 1 ≤ j ≤ s. and the index set of the variables v_{j}) and the bounds (4.19) and (4.21) about the L_{2}-norm of the functions ū_{j}(·|·) and ḡ(·|·).

By rewriting the integrand in formula (4.17) with the help of the identities u_{j}(·) = ū_{j}(·|·)U_{j}(·), 1 ≤ j ≤ s and g(·) = ḡ(·|·)G(·) we can rewrite the expression I in formula (4.17) with the help of the above introduced quantities as

I = ∫ Z(x_{r}, n + 1 ≤ r ≤ n + m) ∏_{j=1}^{s} U_{j}(x_{q}, q ∈ A_{j}^{(2)})G(x_{n+1}, …, x_{n+m}) ∏_{i=n+1}^{n+m} μ( dx_{i}).

Beside this, the L_{2}-norms of the functions

∏_{j=1}^{s} U_{j}(x_{q}, q ∈ A_{j}^{(2)}) and G(x_{n+1}, …, x_{n+m})

are bounded by 1 and D_{2}R^{−1} respectively by relations (4.18) and (4.20), and the function Z(·) satisfies inequality (4.22). These inequalities together with the Schwarz inequality imply that I ≤ D_{1}D_{2}R^{s−2}, i.e. relation (4.17) holds. The proof of the Main Inequality is completed.

5. On Latała’s conjecture.

In this section I discuss Latała’s conjecture. I show, by working out the details of the arguments leading to this conjecture that it is equivalent to an estimate about the expected value of the supremum of certain random multilinear forms. The original form of this conjecture contains an estimate about the moments of certain Gaussian polynomials. The would-be proof applies an inductive argument with respect to the order of the polynomials we consider. It is known that the conjecture holds for polynomials of order 2. I shall consider polynomials of order 3 whose study also reveals very much about the general situation. First I formulate Latała’s conjecture for polynomials of order three in an explicit form.

Latała’s conjecture for Gaussian random polynomials of order 3. Let us fix a large positive integer M and consider a random polynomial of the form

Z = ∑_{i,j,k} a(i, j, k)ξ_{i}η_{j}ζ_{k},  (5.1)

where all random variables ξ_{i}, η_{j} and ζ_{k} have standard normal distribution, and they are independent of each other. Let the coefficients a(i, j, k) of the polynomial Z in formula (5.1) satisfy the following inequalities depending on the fixed parameter M:

∑_{i,j,k} a(i, j, k)u(i, j, k) ≤ 1 if ∑_{i,j,k} u^{2}(i, j, k) ≤ 1,  (5.2)

∑_{i,j,k} a(i, j, k)u(i, j)v(k) ≤ M^{−1/2} if ∑_{i,j} u^{2}(i, j) ≤ 1 and ∑_{k} v^{2}(k) ≤ 1,
∑_{i,j,k} a(i, j, k)u(i, k)v(j) ≤ M^{−1/2} if ∑_{i,k} u^{2}(i, k) ≤ 1 and ∑_{j} v^{2}(j) ≤ 1,
∑_{i,j,k} a(i, j, k)u(j, k)v(i) ≤ M^{−1/2} if ∑_{j,k} u^{2}(j, k) ≤ 1 and ∑_{i} v^{2}(i) ≤ 1,

(5.3)

and

∑_{i,j,k}a(i, j, k)u(i)v(j)w(k) ≤ M^{−1}
if ∑_{i} u^{2}(i) ≤ 1, ∑_{j} v^{2}(j) ≤ 1 and ∑_{k} w^{2}(k) ≤ 1.  (5.4)

Then the random polynomial Z satisfies the inequality

EZ^{2M} ≤ C^{M}M^{M}  (5.5)

with some universal constant C > 0.

In the calculation of EZ^{2M} it is useful to consider first its conditional expectation under the condition that the value of all random variables ξ_{i} are prescribed. This conditional expectation has a simple form which can be well bounded because of the independence of the variables ξ_{i}, η_{j} and ζ_{k}. We get

E(Z^{2M}|ξ_{i} = x_{i}) = E (∑_{i,j,k} a(i, j, k)x_{i}η_{j}ζ_{k})^{2M} = E (∑_{j,k} A(j, k|x)η_{j}ζ_{k})^{2M} , (5.6)

where

A_{i}(j, k|x) = A_{i}(j, k|x_{1}, x_{2}, …) = ∑_{i} a(i, j, k)x_{i}.

The moment estimates known for Gaussian polynomials of order two enables us to bound the expression in formula (5.6). These estimates depend on the Hilbert–Schmidt norm D_{1}(x) and usual norm D_{2}(x) of the matrix A(j, k|x) appearing in formula (5.6). To get a formula more appropriate for our investigations I give the value of these quantities by means of the the following variational principle.

D_{1}(x) = sup ; v(j,k): ∑ v^{2}(j,k)≤1 ∑_{j,k} A(j, k|x)v(j, k),

and

D_{2}(x) = sup ; (v(j), w(k)): ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 ∑_{j,k} A(j, k|x)v(j)w(k).

In such a way we get the following estimate.

\begin{matrix}E(Z^{2M}|ξ_{i} = x_{i}) = E (∑_{j,k} A(j, k|x)η_{j}ζ_{k})^{2M} \\ ≤ C^{M}M^{M} ( sup_{\begin{matrix}v(j,k) \\ ∑ v^{2}(j,k)≤1 \end{matrix}} ∑_{j,k} A(j, k|x)v(j, k))^{2M} \\ + C^{M}M^{2M} (  sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 \end{matrix}} ∑_{j,k} A(j, k|x)v(j)w(k))^{2M} . \end{matrix}  (5.7)

Taking expectation in inequality (5.7) we get that

E(Z^{2M}) ≤ C^{M}M^{M}E ( sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} ∑_{j,k} A(j, k|ξ_{i})v(j, k))^{2M}

+ C^{M}M^{2M}E ( sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 \end{matrix}} ∑_{j,k} A(j, k|ξ_{i})v(j)w(k))^{2M} .

The last inequality can be rewritten in the form

E(Z^{2M}) ≤ C^{M}M^{M}E ( sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} ∑_{i} B_{i,1}(u(j, k))ξ_{i})^{2M}

+ C^{M}M^{2M}E ( sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 \end{matrix}} ∑_{i} B_{i,2}(v(j), w(k))ξ_{i})^{2M} .

with

B_{i,1}(u(j, k), j, k = 1, 2, …,) = ∑_{j,k} a(i, j, k)u(j, k),

and

B_{i,2}(v(j), w(k), j, k = 1, 2, …) = ∑_{j,k} a(i, j, k)v(j)w(k),

or by introducing the notations u = (u(j, k), j, k = 1, 2, …), v = (v(j), j = 1, 2, …) and w = (v(k), k = 1, 2, …) together with the Gaussian random variables

\begin{aligned}X(u) & = ∑_{i,j,k} a(i, j, k)u(j, k)ξ_{i} = ∑_{i} B_{i,1}(u(j, k), j, k = 1, 2, …)ξ_{i}, \\ Y (v, w) & = ∑_{i,j,k} a(i, j, k)v(j)w(k)ξ_{i} = ∑_{i} B_{i,2}(v(j), w(k), j, k = 1, 2, …)ξ_{i}, \end{aligned} (5.8)

this inequality can be written in the form

\begin{aligned}E(Z^{2M}) & ≤ C^{M}M^{M}E ( sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} X(u))^{2M} \\ & + C^{M}M^{2M}E (  sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 \end{matrix}} Y (v, w))^{2M} . \end{aligned}  (5.9)

The right-hand side of (5.9) can be bounded by means of some concentration theorem type inequalities about the supremum of a Gaussian process. Ledoux has a result about the supremum of Gaussian processes (Theorem 7.1 in the book [6] The Concentration of Measure Phenomenon) which states that the supremum of a Gaussian process U(t), EU(t) = 0, t ∈ T, takes a value larger than E sup_{t∈T} U(t) with relatively small prob-

ability. More explicitly, it states that P (sup_{t∈T} U(t) ≥ E sup_{t∈T} U(t) + x) ≤ C_{1}e^{−C_{2}x^{2}/λ}, where λ = sup_{t∈T} EU^{2}(t). Some calculation with the help of this inequality yields the

estimates

E ( sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} X(u))^{2M} ≤ D^{M} (E | sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} X(u)|)^{2M}

+ D^{M} sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} EX(u)^{2M}

≤ D^{M} (E | sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} X(u)|)^{2M} + D^{M}M^{M} sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} (EX(u)^{2})^{M}

and

E ( sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 \end{matrix}} Y (v, w))^{2M}

≤ D^{M} (E | sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 \end{matrix}} Y (v, w)|)^{2M}

+ D^{M} sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, w^{2}(k)≤1 \end{matrix}} EY (v, w)^{2M}

≤ D^{M} (E | sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w(k)^{2}≤1 \end{matrix}} Y (v, w)|)^{2M}

+ D^{M}M^{M} sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w^{2}(k)≤1 \end{matrix}} (EY (v, w)^{2})^{M} .

The content of the above inequalities is that to get a good estimate on the high moments of the supremum of a Gaussian process with expectation zero it is enough to have a good estimate on the expectation of the absolute value of this supremum and on the moments of the single random variables in this stochastic process. The latter terms can be expressed by means of the variance of these random variables.

A relatively simple calculation by means of the Schwarz inequality shows that under conditions (5.3) and (5.4) the inequalities EX(u)^{2} ≤ M^{−1}, and EY (v, w)^{2} ≤ M^{−2} hold if ∑ u(j, k)^{2} ≤ 1, ∑ v^{2}(j) ≤ 1, and ∑ w^{2}(k) ≤ 1. Some calculation also shows that under the condition (5.2) the inequality

E | sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} X(u)| ≤ (E ( sup_{\begin{matrix}u(j,k) \\ ∑ u^{2}(j,k)≤1 \end{matrix}} X(u))^{2})^{1/2} ≤ C  (5.10)

holds. The second term in (5.10) can be well estimated, because the supremum of the random variables X(u) can be explicitly calculated for all fixed random vectors ξ_{i}, i = 1, 2, …, and after this, the middle term in (5.10) can be well bounded with the help of relation (5.2) because of the orthogonality of the random variables ξ_{i}.

Because of the above inequalities to show that relation (5.5) holds under conditions (5.2), (5.3) and (5.4) it would be sufficient to prove the inequality

E |  sup_{\begin{matrix}v(j), w(k) \\ ∑ v^{2}(j)≤1, ∑ w(k)^{2}≤1 \end{matrix}} Y (v, w)| ≤ \frac{C}{\sqrt{M}} .  (5.11)

under the above conditions with the random variables Y (v, w) introduced in (5.8) and some universal constant C < ∞. This would mean that Latała’s conjecture holds for Gaussian polynomials of order 3.

A more careful analysis would even show that the validity of relation (5.11) under conditions (5.2), (5.3) and (5.4) is equivalent to Latała’s conjecture. This argument can be adapted to the case of general parameter k, and it yields that Latała’s conjecture is equivalent to Theorem 2 of his paper [5].

More generally, it can be said that even if we cannot prove inequality (5.11), the bound we can give for the high moments of a random polynomial Z defined in (5.1) depends on what kind of estimate we can prove for the expression in (5.11). But the estimation of such an expression is a hard problem. The analogous problem in formula (5.10) was much simpler.

The estimation of the left-hand side of (5.11) is much harder than that of formula (5.10), because in this case the supremum of random trilinear forms had to be considered (and not the supremum of random bilinear forms as in formula (5.10)). Latała tried to get a good estimate for such an expression by means of a good bound on a quantity N(X, ρ_{α}, ε) which was introduced also in Section 2 of this paper. But his proof of the estimate for N(·) contained an error which seems to be serious. I do not know whether the estimation of the quantity N(X, ρ_{α}, ε) is the right way to give a good bound on the expression in formula (5.11). But the proof of such an estimate demands a deeper analysis than the method of paper [5]. Such an analysis should exploit the finer structure of the model we consider.

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