Moments of Gaussian chaoses in Banach spaces

We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including L_q spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in L_q spaces based on exponential random variables.


Introduction
Multivariate polynomials in Gaussian variables have been extensively studied at least since the work of Wiener in the 1930s. They have found numerous applications in the theory of stochastic integration and Malliavin calculus [12,22,23], functional analysis [11], limit theory for U -statistics [9] or long-range dependent processes [29], random graph theory [12], and more recently computer science [7,14,19,24]. While early results considered mostly polynomials with real coefficients, their vector-valued counterparts also appear naturally, e.g., in the context of stochastic integration in Banach spaces [20], in the study of weak limits of U-processes [9], as tools in characterization of various geometric properties of Banach spaces [11,25,26] or in the analysis of empirical covariance operators [1,30]. Apart from applications, the theory of Gaussian polynomials has been studied for its rich intrinsic structure, with interesting interplay of analytic, probabilistic, algebraic and combinatorial phenomena, leading to many challenging problems. For a comprehensive presentation of diverse aspects of the theory we refer to the monographs [9,11,12,17].
An important aspect of the study of Gaussian polynomials is the order of their tail decay and growth of moments. In the real valued case the first estimates concerning this question, related to the hypercontractivity of the Ornstein-Uhlenbeck semigroup, were obtained by Nelson [21]. For homogeneous tetrahedral (i.e., affine in each variable) forms of arbitrary fixed degree two-sided estimates on the tails and moments were obtained in [15] (in particular generalizing the well-known Hanson-Wright inequality for quadratic forms). In [4] it was shown that the results of [15] in fact allow to obtain such estimates for all polynomials of degree bounded from above. Two-sided estimates for polynomials with values in a Banach space have been obtained independently by Borell [6], Ledoux [16], Arcones-Giné [5]. They are expressed in terms of suprema of certain empirical processes (see formula (1.5) below), which in general may be difficult to estimate (even in the real valued case).
In a recent paper [2] we considered Gaussian quadratic forms with coefficients in a Banach space and obtained upper bounds on their tails and moments, expressed in terms of quantities which are easier to deal with. In the real valued case our estimates reduce to the Hanson-Wright inequality, and for a large class of Banach-spaces (related to Pisier's Gaussian property α and containing all type 2 spaces) they may be reversed. In particular for L q spaces with 1 ≤ q < ∞ they yield two-sided estimates expressed in terms of deterministic quantities. In the present work we generalize these estimates to polynomials of arbitrary degree.
Before presenting our main theorems (which requires an introduction of a rather involved notation) let us describe the setting and discuss in more detail some of the results mentioned above.
To this aim consider a Banach space (F, · ). A (homogeneous, tetrahedral) F -valued Gaussian chaos of order d is a random variable defined as S = 1≤i1<i2...<i d ≤n a i1,...,i d g i1 · · · g i d , where a i1,...,i d ∈ F and g 1 , . . . , g n are i.i.d. standard Gaussian variables. As explained above the goal of this paper is to derive estimates on moments (defined as S p := (E S p ) 1/p ) and tails of S, more precisely to establish upper bounds which for some classes of Banach spaces, including L q spaces, can be reversed (up to constants depending only on d and the Banach space, but not on n or a i1,...,i d ). We restrict to random variables of the form (1.1), however it turns out that estimates on their moments will in fact allow to deduce moment and tail bounds for arbitrary polynomials in Gaussian random variables as well as for homogeneous tetrahedral polynomials in i.i.d. symmetric exponential random variables. In the sequel we will focus on decoupled chaoses i1 · · · g (d) where (g (k) i ) i,k≥1 are independent N (0, 1) random variables -under natural symmetry assumptions, moments and tails of S, S are comparable up to constants depending only on d (cf. Theorem A.9 in the Appendix). Moreover, as we will see in Proposition 2.11, estimating moments of general polynomials in i.i.d. standard Gaussian variables can be reduced to estimating moments of variables of the form (1.2). For d = 1 and any p ≥ 1 one has the following well-known estimate (cf. Lemma A.5) where, B n 2 is the unit standard Euclidean ball in R n and ∼ stands for a comparison up to universal multiplicative constants.

Moments of Gaussian chaoses in Banach spacess
An iteration of the above inequality yields for chaoses of order 2, a ij x i y j , (1.4) where in the above formula and in the whole paper, (g j ) j≥1 is an independent copy of (g i ) i≥1 .
For chaoses of higher order one gets an estimate n i1,...,i d =1 a i1,...,i d g , (1.5) where the supremum is taken over x (1) , . . . , x (n) from the Euclidean unit ball and ∼ a stands for comparison up to constants depending only on the parameter a. To the best of our knowledge the above inequality was for the first time established in [6] and subsequently reproved in various context by several authors [5,16,17].
The estimate (1.5) gives precise dependence on p, but unfortunately is expressed in terms of expected suprema of certain stochastic processes, which are hard to estimate. In many situations this precludes effective applications. Let us note that even for d = 1, the estimate (1.3) involves the expectation of a norm of a Gaussian random vector. Estimating such a quantity in general Banach spaces is a difficult task, which requires investigating the geometry of the unit ball of the dual of F (as described by the celebrated majorizing measure theorem due to Fernique and Talagrand). Therefore, in general one cannot hope to get rid of certain expectations in the estimates for moments. Nevertheless, in some classes of Banach spaces (such as, e.g., Hilbert spaces, or more generally type 2 spaces) expectations of Gaussian chaoses can be easily estimated. The difficult part (also for d = 2 and mentioned class of Banach spaces) is to estimate the terms in (1.4) and (1.5) which involve additional suprema over products of unit balls.
Even for d = 2 and a Hilbert space, the term E sup x∈B n 2 i,j a ij g i x j can be equivalently rewritten as the expected operator norm of a certain random matrix. Such quantities are known to be hard to estimate. Therefore, it is natural to seek inequalities which are expressed in terms of deterministic quantities and expectations of some F -valued polynomial chaoses, but do not involve expectations of additional suprema of such polynomials. This was the motivation behind the article [2], concerning the case d = 2 and containing the following bound, valid for p ≥ 1 ([2, Theorem 4]), a ij x i y j , (1.6) where (with a slight abuse of notation) we denote B n 2 2 = {(x ij ) n i,j=1 : Let us point out that even though the inequalities we have presented so far as well as those we are about to discuss in the subsequent part of the article are formulated for general, possibly infinite dimensional, Banach spaces, the random variables involved take values in finite dimensional subspaces spanned by the coefficients of the polynomials in question. Therefore, as long as the constants in the inequalities are universal or do not depend on the particular subspace but just on some numerical characteristic of the space (e.g., the type constant), there is no loss in generality in assuming that the space F is finite dimensional. In particular one can always assume without loss of generality that F as a linear space equals R m for some positive integer m. This phenomenon is well known in the local theory of Banach spaces.
It can be shown that in general inequality (1.6) cannot be reversed. However, it turns out to be two-sided in a certain class of Banach spaces containing L q spaces (see Section 2.1 below). This observation gives rise to the question of obtaining similar results for arbitrary d. Building on ideas and techniques developed in [15] we are able to give an answer to it. In our main result, Theorem 2.1, we provide an upper bound on moments of decoupled chaoses of order d, which generalizes (1.6). We also obtain lower bounds, which we conjecture to be in fact two-sided (see Conjecture 2.2), and in Section 2.1 we identify a large class of Banach spaces for which our upper and lower bounds do match.
Let us briefly comment on the proof of Theorem 2.1. The lower bound for moments relies on a rather straightforward reduction to the real-valued case, treated in [15]. The much more involved proof of the upper bound is based on an inductive approach. The inequality (1.6) serves as the base of induction, while (1.3) allows to reduce the induction step to an estimate of an expectation of a supremum of a certain canonical Gaussian process, which turns out to be the heart of the problem. In order to obtain such an estimate we apply a variant of the chaining method (see the monograph [28]), which requires bounds on the entropy numbers for the indexing set of the process in its intrinsic metric. In our case this metric is given via a norm on a tensor product of F * and several Euclidean spaces, whereas the indexing set is a Carthesian product of the corresponding unit balls. The bounds on entropy numbers are obtained by a variant of the volumetric argument as well as Sudakov and dual Sudakov minoration leading to expectations of suprema of other Gaussian processes, which can be estimated by using the induction hypotheses. This approach in a sense parallels the one used in [15] for the real-valued case, with (1.6) replacing the classical Hanson-Wright inequality, however it presents some additional difficulties related to the geometry of the unit ball in the space F * . Let us also remark that this approach cannot be used to pass from d = 1, i.e., the inequality (1.3), to d = 2, i.e., the inequality (1.6) (see Remark 3.7). While the proof of (1.6) presented in [2] relies on a similar set of tools (chaining arguments, Gaussian concentration) some of the technical estimates of entropy numbers are obtained differently than in the induction The paper is organized as follows. In the next section we set up the notation and formulate the main results, in particular the pivotal bound for moments of homogeneous tetrahedral Gaussian chaoses in an arbitrary Banach space (Theorem 2.1). We also present its consequences: tail and moment estimates for arbitrary Gaussian polynomials, two-sided bounds in special classes of Banach spaces, inequalities for tetrahedral homogeneous forms in i.i.d. symmetric exponential variables. In Section 3, in Theorem 3.1, we formulate a key inequality for the supremum of a certain Gaussian processes and derive certain entropy bounds to be used in its proof, presented in Section 4. In Section 5 we use Theorem 3.1 to prove Theorem 2.1 from which we deduce all the remaining claims of Section 2. The Appendix contains certain basic facts concerning Gaussian processes and Gaussian polynomials used in the proofs. At the end of the article we provide a glossary explaining the notation.

Notation and main results
In this section we introduce the most basic notation used in the article and formulate our main results. Since some additional notation will be introduced as the proofs develop, for the reader's convenience at the end of the article we include a glossary of the most important symbols appearing in the text.
We write [n] for the set {1, . . . , n}. Throughout the article C (resp. C(α)) will denote an absolute constant (resp. a constant which may depend on α) which may differ at each occurrence. By A we typically denote a finite multi-indexed matrix (a i1,...,i d ) 1≤i1,...,i d ≤n of order d with values in a Banach space (F, · ). If , then we define i I := (i j ) j∈I . To simplify the notation we will also often treat i I as a stand-alone multi-index, with the meaning that each i j , j ∈ I runs trough [n].
We will also often suppress the range of summation. Unless stated otherwise the sums i or i1,...,i d should be understood as summation over i 1 , . . . , i d ∈ [n], whereas i I should be understood as summation over i j1 , . . . , i j k ∈ [n] where I = {i 1 , . . . , j k }. The parameter d will not be stated explicitly but will be clear from the context. In particular when we write x = (x i I ) i I , it is implicitly assumed that the multi-index i I ranges over We note that all the multi-linear forms we consider are given by finite sums. Standard arguments allow to extend our inequalities to the case of infinite multiple series, but we do not pursue this direction.
In what follows we will often identify the space (R n ) ⊗d of d-indexed matrices with the space R n d . In particular B n d 2 will stand for the unit Euclidean ball in (R n ) ⊗d , i.e., . If I is a finite set then |I| stands for its cardinality and by P(I) we denote the family of (unordered) partitions of I into nonempty, pairwise disjoint sets. Note that if I = ∅ then P(I) consists only of the empty partition ∅.
With a slight abuse of notation we write (P, P ) ∈ P(I) if P ∪ P ∈ P(I) and P ∩ P = ∅. Let P = {I 1 , . . . , I k }, P = {J 1 , . . . , J m } be such that (P, P ) ∈ P( [d]).Then we define  We do not exclude the situation that P or P is an empty partition. If P = ∅, then A P | P = |||A||| P is defined in non-probabilistic terms. Another case when A P | P = |||A||| P is when P consists of singletons only.
In particular for d = 3 we have (note that to shorten the notation we suppress some brackets and write e.g. |||A||| {2} The main result is the following moment estimate of the variable S . The lower bound in (2.3) motivates the following conjecture (we leave it to the reader to verify that in general Banach spaces it is impossible to reverse the upper bound even for d = 2).
i g where (we recall) C is a numerical constant and Remark 2.4. Unfortunately we are able to show (2.4) only for d = 2 and with an additional factor ln p (cf. [2]). It is likely that by a modification of our proof one can show (2.4) for arbitrary d with an additional factor (ln p) C (d) .
By a standard application of Chebyshev's and Paley-Zygmund inequalities, Theorem 2.1 can be expressed in terms of tails.
and for any t ≥ 0, In view of (1.5) and [15] it is clear that to prove Theorem 2.1 one needs to estimate suprema of some Gaussian processes. The next statement is the key element of the proof of the upper bound in (2.3). Theorem 2.6. Under the assumptions of Theorem 2.1 we have for any p ≥ 1, We postpone proofs of the above results until Section 5 and discuss now some of their consequences.

Two-sided estimates in special classes of Banach spaces
We start by introducing a class of Banach spaces for which the estimate (2.3) is two-sided. To this end we restrict our attention to Banach spaces (F, · ) which satisfy the following condition: there exists a constant K = K(F ) such that for any n ∈ N and This property appears in the literature under the name Gaussian property (α+) (see [20]) and is closely related to Pisier's contraction property [25]. It has found applications, e.g., in the theory of stochastic integration in Banach spaces. We refer to [11,Chapter 7] for a thorough discussion and examples, mentioning only that (2.6) holds for Banach spaces of type 2, and for Banach lattices (2.6) is equivalent to finite cotype. Remark 2.7. By considering n = 1 it is easy to see that K ≥ π/2 > 1.
A simple inductive argument and (2.6) yield that for any d, n ∈ N and any F -valued where we recall that i = (i 1 , . . . , i d ) and each i 1 , . . . , i d runs trough [n]. It turns out that under the condition (2.6) our bound (2.3) is actually two-sided.
The following corollary is an obvious consequence of Proposition 2.8 and Theorems 2.1, 2.5.

Corollary 2.9. For any Banach space
Thanks to infinite divisibility of Gaussian variables, the above corollary can be in fact generalized to arbitrary polynomials in Gaussian variables, as stated in the following theorem.
and for all t > 0, The above theorem is an easy consequence of results for homogeneous decoupled chaoses and the following proposition, the proof of which (as well as the proof of the theorem) will be presented in Section 5. Proposition 2.11. Let F be a Banach space, G a standard Gaussian vector in R n and f : R n → F be a polynomial of degree D. Then for p ≥ 1, Remark 2.12. Let us stress that Proposition 2.11 as well as inequalities (2.8) and (2.9) of Theorem 2.10 hold in arbitrary Banach spaces. The assumption (2.6) is needed for inequalities (2.10) and (2.11). A positive answer to Conjecture 2.2 would allow to eliminate this assumption and remove the constant K from the inequalities.

L q spaces
It turns out that L q spaces satisfy (2.6) and as a result upper and lower bounds in (2.3) are comparable. Moreover, as is shown in Lemma 2.14 below, in this case one may express all the parameters without any expectations. For the sake of brevity, we will focus on moment estimates, clearly tail bounds follow from them by standard arguments (cf. the proof of Theorem 2.5).
The following lemma asserts that for general J the corresponding two norms are comparable.
Proof. By Jensen's inequality and Corollary A.7 we get On the other hand Theorem A.1 (applied with p = 1) and Corollary A.7 yield Proof. This is an obvious consequence of Theorem 2.1, Corollary 2.9, Proposition 2.13 and Lemma 2.14.
Using Proposition 2.11 we can extend the above result to general polynomials.
Theorem 2.16. Let G be a standard Gaussian vector in R n and let f : where the coefficients a ijk , b ij , c i , d take values in a Banach space and the matrices (a ijk ) ijk , (b ij ) ij are symmetric. Then one checks that ,

Exponential variables
Theorem 2.15 together with Lemma A.8 allows us to obtain inequalities for chaoses based on i.i.d standard symmetric exponential random variables (i.e., variables with Similarly as in the previous section we concentrate only on the moment estimates. One can take C −1 (d)q 1/2−d in the lower bound and C(d)q 2d−1/2 in the upper bound.

Moments of Gaussian chaoses in Banach spacess
The proof of Proposition 2.18 is postponed until Section 5.

Reformulation of Theorem 2.6 and entropy estimates
Let us rewrite Theorem 2.6 in a different language. As explained in the introduction, since the variables we consider take values in the finite dimensional subspace spanned by the coefficients a i , we may assume without loss of generality that F = R m for some finite m and a i1,...,i d = (a i1,...,i d ,i d+1 ) i d+1 ≤m . For this reason from now on the multi-index i will take values in [n] d × [m] and all summations over i should be understood as summations over this set. Accordingly, the matrix A will be treated as a (d + 1)-indexed matrix with real coefficients. Let T = B F * be the unit ball in the dual space F * (where duality is realized on R m through the standard inner product). In the sequel we will therefore assume that T is a fixed nonempty symmetric bounded subset of R m .
To make the notation more compact we define The next statement is a reformulation of Theorem 2.6 in the introduced setup. The proof of it will be presented in Section 4. x (k) To estimate the supremum of a centered Gaussian process (G v ) v∈V one needs to study the distance on V given by d(v, v ) := (E|G v − G v | 2 ) 1/2 (we refer to the monograph [28] for an extensive presentation of chaining techniques related to estimates of suprema of stochastic processes). In the case of the Gaussian process from (3.2) this distance is ((x (2) , . . . , x (d) , t), (y (2) , . . . , y (d) , t )) We will now provide estimates for the entropy numbers N (U, ρ A , ε) for ε > 0 and U ⊂ (B n 2 ) d−1 × T (recall that N (S, ρ, ε) is the minimal number of closed balls with diameter ε in metric ρ that cover the set S). To this end let us introduce some new notation. From now on G n = (g 1 , . . . , g n ) and G by (recall that we assume symmetry of where Let us note that V U I (β A ) depends on the set U only through its projection on the first d − 1 coordinates.
We have x (k) . (3.8) Observe that by the classical Sudakov minoration (see Theorem A.2), for any We define a measure µ d ε,T on R (d−1)n × T by the formula where γ n,t is the distribution of tG n = t(g 1 , . . . , g n ). Clearly, For any (y (2) , . . . , y (d) ) ∈ U there exists t ∈ T y (k) ,ε such that where in the third inequality we used (3.8). Thus, where the last inequality follows by Lemma 3.2 applied to the norm α A (· ⊗ t) + εβ A (·).
and for any δ > 0, Proof. It suffices to show (3.10), since it easily implies (3.11). Consider first ε ≤ 8. ε)). Therefore, by Lemma 3.3 (applied with ε/16) we have for any (x, t) ∈ U , where ∆ is equal to the diameter of the set (B n 2 ) d−1 × T in the metric ρ A . Unfortunately the entropy bound derived in Corollary 3.4 involves a nonintegrable term δ −1 . The remaining part of the proof of Theorem 3.1 is devoted to improving on Dudley's bound.
Proof. We will estimate the quantities W U d (α A , ε) and V U d (β A , ε) appearing in Corollary 3.4.
Since U ⊂ (B n 2 ) d−1 × T , Jensen's inequality yields for I ⊂ {2, . . . , d}, By estimating a little more accurately in the second inequality in (3.13) we obtain for (3.14) Observe that (3.14) is not true for d = 2 (cf. Remark 3.7). Let us now pass to the quantity V U d (β A , ε). The definition of V U I and the inclusion Hence the assertion is a simple consequence of Corollary 3.4.
Remark 3.7. Proposition 3.6 is not true for d = 2. The problem arises in (3.14) -for d = 2 there does not exist P ∈ P( [d] \ {l}) such that |P| = d − 2. This is the main reason why proofs for chaoses of order d = 2 (cf. [2]) have a different nature than for higher order chaoses.

Proof of Theorem 3.1
We will prove Theorem 3.1 by induction on d (recall that the matrix A has order d + 1). To this end we need to amplify the induction hypothesis.
Clearly it is enough to prove Theorem 4.1 for finite sets U . Observe that thus Theorem 4.1 implies Theorem 3.1. We will prove (4.1) by induction on d, but first we will show several consequences of the theorem. In the next three lemmas, we shall assume that Theorem 4.1 (and thus also Theorem 3.1) holds for all matrices of order smaller than d + 1.
Proof. Note that Up to a permutation of the indexes we have two possibilities ij a i g ij where D M P | P is defined in the same manner as A P | P (see (3.1)) but the supremum is taken over the set M instead of T . The second inequality in (4.6) can be justified analogously as (4.4).
Recall also that s k (A(y, I)) ≤ s k+|I| (A), thus we may apply 2 d−1 − d times Proposition 3.6 with ε = 2 −l p −1/2 and find a decomposition U = N1 j=1 U j , N 1 ≤ exp(C(d)2 2l p) such that for each j and I ⊂ {2, . . . , d} with |I| < d − 2, is an n × m matrix). We define also (as in (3.6)) Recall the definitions (3.5) and (3.7) and note that (denoting by U the projection of U onto the j-th and (d + 1)-th coordinate) where we again used that y k ∈ B n 2 , U ⊂ (B n 2 ) d−1 × T .
Proof. By Lemma 4.2 we get where the diameter of the sets B i in the normα satisfies .
. Selecting arbitrary (y i , t i ) ∈ U i (we can assume that these sets are nonempty) and using Lemma 4.
Without loss of generality we can assume N ≥ 2 and |U i,j | ≤ |U |−1.
We will now show that for r ≥ 2,

Proofs of main results
We return to the notation used Section 2. In particular in this section the multi-index where F * is the dual space and in the second inequality we used Theorem A.6.
The upper bound will be proved by an induction on d. For d = 2 it is shown in [2]. Suppose that d ≥ 3 and the estimate holds for F -valued matrices of order 2, . . . , d − 1. By the induction assumption, we have Choose P = (I 1 . . . , I k ), P = (J 1 , . . . , J m ) and denote J = P . By the definition of A P | P we have  Proof of Theorem 2.5. Let S = a i g  p |P|/2 A P | P := t 1 + t 2 and observe that if t 1 < t 2 then The first inequality of the theorem follows then by adjusting the constants.
On the other hand by the Paley-Zygmund inequality we get for p ≥ 2, where in the last inequality we used Theorem A.1. The inequality follows by a similar substitution as for the upper bound.

Proof of Proposition 2.11 and Theorem 2.10
Let us first note that Proposition 2.11 reduces (2.8) of Theorem 2.10 to the lower estimate given in Theorem 2.1, while (2.10) is reduced to Corollary 2.9. The tail bounds (2.9) and (2.11) can be then obtained by Chebyshev's and Paley-Zygmund inequalities as in the proof of Theorem 2.5. The rest of this section will be therefore devoted to the proof of Proposition 2.11.
The overall strategy of the proof is similar to the one used in [4] to obtain the real valued case of Theorem 2.10. It relies on a reduction of inequalities for general polynomials of degree D to estimates for decoupled chaoses of degree d = 1, . . . , D. To this end we will approximate general polynomials by tetrahedral ones and split the latter into homogeneous parts of different degrees, which can be decoupled. The splitting may at first appear crude but it turns out that up to constants depending on D one can in fact invert the triangle inequality, which is formalized in the following result due to Kwapień (see [13,Lemma 2]). Recall that a multivariate polynomial is called tetrahedral, if it is affine in each variable.
Theorem 5.1. If X = (X 1 , . . . , X n ) where X i are independent symmetric random variables, Q is a multivariate tetrahedral polynomial of degree D with coefficients in a Banach space E and Q d is its homogeneous part of degree d, then for any symmetric convex function Φ : E → R + and any d ∈ {0, 1, . . . , D}, It will be convenient to have the polynomial f represented as a combination of multivariate Hermite polynomials: In what follows, we will use the following notation. For a set I, by I k we will denote the set of all one-to-one sequences of length k with values in I. For an F -valued d-indexed matrix A = (a i1,...,i d ) i1,...,i d ≤n and x ∈ R n d (R n ) ⊗d we will denote Let (W t ) t∈[0,1] be a standard Brownian motion. Consider standard Gaussian random variables g = W 1 and, for any positive integer N , Approximating the multiple stochastic integral leads to where the limit is in L 2 (Ω) (see [12,Theorem 7.3. and formula (7.9)]) and actually the convergence holds in any L p (see [12,Theorem 3.50]). Now, consider n independent copies (W Let also 1,N , . . . , g N,N , g 1,N , . . . , g N,N , . . . , g 1,N , . . . , g (n) N,N ) = (g (i) j,N ) ( N ) ) ⊗d N →∞ −→ h d1 (g (1) ) · · · h dn (g (n) ) in L p (Ω), which together with the triangle inequality implies that for any p > 0, where G = (g (1) , . . . , g (n) ) and we interpret multiplication of an element of F and a real valued d indexed matrix in a natural way. Thus, by Theorem 5.1 and the triangle inequality we obtain

Proof of the bound for chaoses in exponential variables
Proof of Proposition 2.18. Lemma A.8 implies p |P| 2 |||Â||| Lq,P . (5.14) We will now express J⊂  where in the second equality we used the fact that (y (l) together with convexity and homogeneity of the norm By combining the above with (5.14)-(5.16) and comparing the exponents of p we conclude the assertion of the proposition. Step 1. Assume first that J ∩ ( k i=1 I i ) = ∅. Without loss of generality we can assume that 1 ∈ J ∩ I 1 . DenoteÎ 1 = I 1 \ {1} and for any matrix (x (1) Observe that for any f 1 , . . . , f n ∈ L q (X, dµ) the function is convex, we obtain similarly as in Step 1, otherwise .
An iteration of this argument shows that indeed one can assume that M satisfies the implication I m ∩ I l = ∅ ⇒ (|I l | = |I m | = 1, I l = I m ).

A Appendix
In this section we gather technical facts that are used in the proof.
Lemma A.5. Let G be a Gaussian variable in a Banach space (F, · ). Then for any p ≥ 2, where (F * , · * ) is the dual space to (F, · ). that a i1,...,i d = 0 whenever there exist 1 ≤ k < m ≤ d such that i k = i m . Then for any p ≥ 1, Moreover, for any t > 0,