aa r X i v : . [ m a t h . P R ] N ov Moments of the Gaussian Chaos
Joseph Lehec ∗ Abstract
This paper deals with Lata la’s estimation of the moments of Gaussian chaoses. It isshown that his argument can be simplified significantly using Talagrand’s generic chaining.Published in
S´eminaire de Probabilit´es XLIII , Lecture Notes in Math. 2006, Springer,2011.
In the article [3], Lata la obtains an upper bound on the moments of the Gaussian chaos Y = X a n ,...,n d g n · · · g n d , where g , g , . . . is a sequence of independant standard Gaussian random variables and the a n ,...,n d are real numbers. His bound his sharp up to constants depending only on the order d of the chaos. The purpose of the present paper is to give another proof of Lata la’s result.Observe that the case d = 1 is easy, since (cid:0) | X a i g i | p (cid:1) /p = ( X a i ) / (cid:0) E | g | p (cid:1) /p ∼ √ p ( X a i ) / . When d = 2, Lata la recovers a result by Hanson and Wright [2] which involves the operatorand the Hilbert-Schmidt norms of the matrix a = ( a ij ) (cid:0) E | X a ij g i g j | p (cid:1) /p ∼ √ p k a k HS + p k a k op . It is known (see [5]) that the moments of the decoupled chaos˜ Y = X a n ,...,n d g n , · · · g n d ,d where ( g i,j ) is a family of standard independant Gaussian variables, are comparable to thoseof Y wih constants depending only on d . Using this fact and reasonning by induction onthe order d of the chaos, Lata la shows that the problem boils down to the estimation of thesupremum of a complicated Gaussian process. Given a set T and a Gaussian process ( X t ) t ∈ T ,estimating E sup T X t amounts to studying the metric space ( T, d) where d is given by theformula d( s, t ) = (cid:0) E( X s − X t ) (cid:1) / . ∗ CEREMADE (UMR CNRS 7534) Universit´e Paris-Dauphine. [email protected] X t = 0for all t ∈ T ) then there exists a universal constant C such thatE sup T t ≤ C Z ∞ p log N ( T, d , ǫ ) d ǫ, where the entropy number N ( T, d , ǫ ) is the smallest number of balls (for the distance d)of radius ǫ needed to cover T . Let us refer to Fernique [1] for a proof of this inequalityand several applications. However, Dudley’s inequality is not sharp: there exist Gaussianprocesses for which the integral is much larger than the expectation of the sup. Unfortunately,the phenomenon occurs here. Lata la is able to give precise bounds for the entropy numbers,but Dudley’s integral does not give the correct order of magnitude. Something finer is needed.The precise estimate of the supremum of a Gaussian process in terms of metric entropywas found by Talagrand. This was the famous Majorizing Measure Theorem [6], which isnow called
Generic chaining , see the book [7]. Lata la did not manage to use Talagrand’stheory, and his proof contains a lot of tricky entropy estimates to beat the Dudley bound.We find this part of his paper very hard to read, and our purpose is to short-circuit it usingTalagrand’s generic chaining.Lastly, let us mention that we disagree with P. Major who released an article on arXiv inwhich he claims that Lata la’s proof is incorrect. The present paper is all about understandingLata la’s work, not correcting it. L norms To avoid heavy multi-indices notations, it is convenient to use tensor products. If X and Y are finite dimensional normed spaces, the notation X ⊗ ǫ Y stands for the injective tensorproduct of X and Y , so that X ⊗ ǫ Y is isometric to L ( X ∗ , Y ) equipped with the operatornorm. If X and Y are Euclidean spaces, we denote by X ⊗ Y their Euclidean tensor product.Moreover, in this case we identify X and X ∗ , so that X ⊗ Y is isometric to L ( X, Y ) equippedwith the Hilbert-Schmidt norm.Throughout the article [ d ] denotes the set { , . . . , d } . Let E , . . . , E d be Euclidean spaces.Given a non-empty subset I = { i , . . . , i p } of [ d ], we let E I = E i ⊗ · · · ⊗ E i p . Also, by convention E ∅ = R . The notation k·k I stands for the norm of E I and B I = { x ∈ E I ; k x k I ≤ } for its unit ball. Let A ∈ E [ d ] and P = { I , . . . , I k } be a partition of [ d ], we let k A k P be thenorm of A as an element of the space E I ⊗ ǫ · · · ⊗ ǫ E I k . When d = 2 for instance, the tensor A can be seen as a linear map from E to E , then k A k { }{ } and k A k { , } are the operator and Hilbert-Schmidt norms of A , respectively. Let us http://arxiv.org/abs/0803.1453 d = 3 and that E = E = E = L ( µ ) for some measure µ . Then for any f ∈ E ⊗ E ⊗ E which we identify L ( µ ⊗ ), we have k f k { }{ , } = sup (cid:0)Z f ( x, y, z ) u ( x ) v ( y, z ) d µ ( x )d µ ( y )d µ ( z ) (cid:1) , where the sup is taken over all u, v having L norms at most 1. Going back to the generalsetting, let us define for a non-empty subset I of [ d ] and an element x ∈ E I the contraction h A, x i to be the image of x by A , when A is seen as an element of L ( E I , E [ d ] \ I ). Then forevery partition P = { I , . . . , I k } we have k A k P = sup (cid:8) h A, x ⊗ · · · ⊗ x k i ; x j ∈ B I j (cid:9) . If Q = { J , . . . , J l } is a finer partition than P (this means that any element of Q is containedin an element of P ) then { x ⊗ · · · ⊗ x l , x j ∈ B J j } ⊂ { y ⊗ · · · ⊗ y k , y j ∈ B I j } , hence k A k Q ≤ k A k P . In particular, k A k { }···{ d } ≤ k A k P ≤ k A k [ d ] . If P is a partition of [ d ], its cardinality card P is the number of subsets of [ d ] in P . Let E , . . . , E d be Euclidean spaces and A ∈ E [ d ] . Let X , . . . , X d be independant random vectorssuch that for all i , the vector X i is a standard Gaussian vector of E i . The (real) randomvariable Z = h A, X ⊗ · · · ⊗ X d i is called decoupled Gaussian chaos of order d . Here is the main result of Lata la. Theorem 1.
There exists a constant α d depending only on d such that for all p ≥ (cid:0) E | Z | p (cid:1) /p ≤ α d X P p card P k A k P , the sum running over all partitions P of [ d ] . The following theorem and corollary are intermediate results from which the previoustheorem shall follow; however we believe they are of independent interest.
Theorem 2.
Let F , . . . , F k +1 be Euclidean spaces, let A ∈ F [ k +1] and X be a standardGaussian vector on F k +1 , recall that h A, X i ∈ F ⊗ · · · ⊗ F k . Then for all τ ∈ (0 , : E kh A, X ik { }···{ k } ≤ β k X P τ k − card P k A k P , where the sum runs over all partitions P of [ k + 1] and the constant β k depends only on k . Corollary 3.
Under the same hypothesis, we have for all p ≥ (cid:16) E kh A, X ik p { }···{ k } (cid:17) /p ≤ δ k X P p card P− k k A k P . roof. Let f : x ∈ F k +1
7→ kh
A, x ik { }···{ k } . Let us use the concentration property of theGaussian measure, which asserts that Lipschitz functions are close to their means with highprobability. More precisely, letting m = E f ( X ), we have for all p ≥ (cid:0) E | f ( X ) − m | p (cid:1) /p ≤ δ ′ √ p k f k lip , where k f k lip is the Lipschitz constant of f and δ ′ is a universal constant. We refer to [4] formore details on this inequality. Noting that k f k lip = sup x ∈ B k +1 kh A, x ik { }···{ k } = k A k { }···{ k +1 } . and using the triangle inequality, we get (cid:0) E | f ( X ) | p (cid:1) /p ≤ E f ( X ) + δ ′ √ p k A k { }···{ k +1 } . The result then follows from the upper bound on E f ( X ) given by Theorem 2 with τ = p − / .Let us prove Theorem 1. We proceed by induction on d . When d = 1, the random variable h A, X i is, in law, equal to the Gaussian variable of variance k A k { } . The p -th moment of thestandard Gaussian variable being of order √ p , we get (cid:0) E |h A, X i| p (cid:1) /p ≤ α √ p k A k { } for some universal α , hence the theorem for d = 1.Assume that the result holds for chaoses of order d −
1. From now on, if I = { i , . . . , i r } is asubset of [ d ] we denote the tensor X i ⊗ · · · ⊗ X i r by X I . Notice that h A, X [ d ] i = (cid:10) h A, X d i , X [ d − (cid:11) and apply the induction assumption to the matrix B = h A, X d i . This yieldsE (cid:0) |h B, X [ d − i| p (cid:12)(cid:12) X d (cid:1) ≤ α pd − (cid:16)X P p card P k B k P (cid:17) p , where the sum runs over all partitions P of [ d − p -th root, weobtain (cid:0) E |h A, X [ d ] i| p (cid:1) /p ≤ α d − E (cid:16)X P p card P kh A, X d ik P (cid:17) p ! /p ≤ α d − X P p card P (cid:16) E kh A, X d ik p P (cid:17) /p , (1)by the triangle inequality. Let P = { I , . . . , I k } be a partition of [ d − F i = E I i for i ∈ [ k ] and F k +1 = E d . The tensor A can be seen as an element of F [ k +1] , let us rename it A ′ when we do so. Corollary 3 gives (cid:0) E kh A ′ , X d ik p { }···{ k } (cid:1) /p ≤ δ k p − k X Q p card Q k A ′ k Q , Q of [ k ]. Going back to the the space E [ d ] , thisinequality translates as (cid:0) E kh A, X d ik p P (cid:1) /p ≤ δ k p − k X Q p card Q k A k Q , (2)and this time the sum runs over partitions Q of [ d ] such that the partition (cid:8) I , . . . , I k , { d } (cid:9) is finer than Q . However, the inequality still holds if we take the sum over all partitions of [ d ]instead. We plug (2) into (1), the numbers p card P cancel out and we get the desired inequalitywith constant α d = α d − X P δ card P , where the sum is taken over all partitions P of [ d − Let F , . . . , F k +1 be Euclidean spaces, let A ∈ F [ k +1] and X be a standard Gaussian vectorof F k +1 . For i ∈ [ k ] let B i be the unit ball of F i , let T = B × · · · × B k . Recall that for x = ( x , . . . , x k ) ∈ T , the notation x [ k ] stands for the tensor x ⊗ · · · ⊗ x k . Note thatE kh A, X ik { }···{ k } = E sup x ∈ T h A, x [ k ] ⊗ X i = E sup x ∈ T (cid:10) h A, x [ k ] i , X (cid:11) . (3)Notice also that ( P x ) x ∈ T = (cid:0)(cid:10) h A, x [ k ] i , X (cid:11)(cid:1) x ∈ T is a Gaussian process. To estimate E sup T P x ,we shall study the metric space ( T, d), whered( x, y ) = (cid:0) E( P x − P y ) (cid:1) / . This distance can be computed explicitly. Indeedd( x, y ) = E (cid:10) h A, x [ k ] − y [ k ] i , X (cid:11) = kh A, x [ k ] − y [ k ] ik { k +1 } . (4)The generic chaining , introduced by Talagrand, will be our main tool. We sketch briefly themain ideas of the theory and refer to Talagrand’s book [7] for details.Let ( T, d) be a metric space. If S is a subset of T we let δ d ( S ) be the diameter of Sδ d ( S ) = sup s,t ∈ S d( s, t ) . Given a sequence (cid:0) A n (cid:1) n ∈ N of partitions of T and an element t ∈ T , we let A n ( t ) be the uniqueelement of A n containing t . Definition 4.
Let γ d ( T ) = inf (cid:16) sup t ∈ T ∞ X n =0 δ d (cid:0) A n ( t ) (cid:1) n/ (cid:17) , where the infimum is over all sequences of partitions (cid:0) A n (cid:1) n ∈ N of T satisfying the cardinalitycondition A = { T } and ∀ n ≥ , card A n ≤ n . (5)5otice that γ d ( T ) ≥ δ d ( T ). In particular, if the metric is not trivial then γ d ( T ) is non-zero. Thus there exists a sequence of partitions ( A n ) n ∈ N satisfying the cardinality conditionand sup t ∈ T ∞ X n =0 δ d (cid:0) A n ( t ) (cid:1) n/ ≤ γ d ( T ) . We recall the all important
Theorem 5 (Majorizing Measure) . There exists a universal constant κ such that for anyGaussian process ( X t ) t ∈ T that is centered (meaning E X t = 0 for all t ∈ T ) we have κ γ d ( T ) ≤ E sup t ∈ T X t ≤ κγ d ( T ) , where the metric d is defined by d( s, t ) = (cid:0) E( X s − X t ) (cid:1) / . Here are two simple lemmas.
Lemma 6.
Let ( T, d) be a metric space. Let a, b ≥ , and ( A n ) n ∈ N be a sequence of partitionsof T satisfying ∀ n ∈ N , card A n ≤ a + b n . Letting γ = sup t ∈ T P ∞ n =0 δ d (cid:0) A n ( t ) (cid:1) n/ , we have γ d ( T ) ≤ ρ (cid:0) √ ab δ d ( T ) + √ b γ (cid:1) , for some universal ρ .Proof. Let p, q be the smallest integers satisfying a ≤ p and b ≤ q . Let B n = (cid:26) { T } if n ≤ p + q A n − q − if n ≥ p + q + 1 . If n ≥ p + q + 1 then p ≤ n − B n ≤ p +2 n − ≤ n . Thus the sequence ( B n ) n ∈ N satisfies (5). On the other hand, for all t ∈ T ∞ X n =0 δ d (cid:0) B n ( t ) (cid:1) n/ = p + q X n =0 δ d ( T )2 n/ + ∞ X n = p δ d (cid:0) A n ( t ) (cid:1) n + q +12 ≤ p + q +12 − √ − δ d ( T ) + 2 q +12 γ. Moreover 2 p ≤ a and 2 q ≤ b , hence the result. Lemma 7.
Let d , . . . , d N be distances defined on T and let d = P d i . Then γ d ( T ) ≤ ρ ′ √ N N X i =1 γ d i ( T ) , where ρ ′ is a universal constant. roof. For all i ∈ [ N ], there exists a sequence ( A in ) n ∈ N of partitions of T satisfying thecardinality condition (5) and sup t ∈ T ∞ X n =0 δ d i (cid:0) A in ( t ) (cid:1) n/ ≤ γ d i ( T ) . Then let A n = { A ∩ · · · ∩ A N , A i ∈ A in } . This clearly defines a sequence of partitions of T , and for all n we havecard A n ≤ N n . (6)On the other hand, for all t ∈ T and i ∈ [ N ] we have A n ( t ) ⊂ A in ( t ), so δ d (cid:0) A n ( t ) (cid:1) ≤ N X i =1 δ d i (cid:0) A n ( t ) (cid:1) ≤ N X i =1 δ d i (cid:0) A in ( t ) (cid:1) . Consequently sup t ∈ T ∞ X n =0 δ d (cid:0) A n ( t ) (cid:1) n/ ≤ N X i =1 γ d i ( T ) . (7)By the previous lemma, equations (6) and (7) yield the result. The proof is by induction on k . When k = 1 the theorem is a consequence of the following:let A ∈ F ⊗ F and X be a standard Gaussian vector on F , thenE kh A, X ik { } ≤ (cid:0) E kh A, X ik { } (cid:1) / = k A k { , } . Assume that k ≥ k ′ < k . Let A ∈ F [ k +1] . Recall thatfor i ∈ [ k ] the unit ball of F i is denoted by B i and the product B × · · · × B k by T . Let I bea non-empty subset of [ k ] and d I be the pseudo-metric on T defined byd I ( x, y ) = kh A, x I − y I ik [ k +1] \ I . (8)By the majorizing measure theorem and the equations (3) and (4), Theorem 2 is equivalentto Theorem 8.
For all τ ∈ (0 , γ d [ k ] ( T ) ≤ β ′ k X P τ k − card P k A k P , with a sum running over all partitions P of [ k + 1] . k . Let τ be a fixed positive realnumber and let d τ be the following metric:d τ = X ∅ ( I ( [ k ] τ k − card I d I . (9)Let us sketch the argument. First we use an entropy estimate and the generic chaining tocompare γ d [ k ] ( T ) and γ d τ ( T ), then we use the induction assumption to estimate the latter.Here is the crucial entropy estimate of Lata la [3, Corollary 2]. Lemma 9.
Let S ⊂ T , let τ ∈ (0 , and ǫ = δ d τ ( S ) + τ k k A k [ k +1] . Then N (cid:0) S, d [ k ] , ǫ (cid:1) ≤ c k τ − , for some constant c k depending only on k . Let us postpone the proof to the last section.Let ( B n ) n ∈ N be a sequence of partitions of T satisfying the cardinality condition (5) andsup t ∈ T ∞ X n =0 δ d τ (cid:0) B n ( t ) (cid:1) n/ ≤ γ d τ ( T ) . (10)Let n ∈ N and B ∈ B n , set τ n = min( τ, − n/ ) and ǫ n = δ d τn ( B ) + τ kn k A k [ k +1] . Observe that τ − n ≤ τ − + 2 n and apply Lemma 9 to B and τ n : N ( B, d [ k ] , ǫ n ) ≤ c k τ − n ≤ c k τ − + c k n . Therefore we can find a partition A B of B whose cardinality is controlled by the numberabove and such that any R ∈ A B satisfies δ d [ k ] ( R ) ≤ ǫ n ≤ δ d τ ( B ) + 2 τ kn k A k [ k +1] . Indeed τ n ≤ τ implies that d τ n ≤ d τ . Then we let A n = ∪{A B ; B ∈ B n } . This clearly definesa sequence of partitions of T which satisfiescard A n ≤ c k τ − + c k n card B n ≤ c k τ − +( c k +1)2 n , (11) δ d [ k ] (cid:0) A n ( t ) (cid:1) ≤ δ d τ (cid:0) B n ( t ) (cid:1) + 2 τ kn k A k [ k +1] , (12)for all n ∈ N and t ∈ T . Recall that τ n = min( τ, − n/ ), an easy computation shows that ∞ X n =0 τ kn n/ ≤ Cτ k − for some universal C . Therefore, for all t ∈ T , we have ∞ X n =0 δ d [ k ] (cid:0) A n ( t ) (cid:1) n/ ≤ ∞ X n =0 (cid:0) δ d τ ( B n ( t )) + τ kn k A k [ k +1] (cid:1) n/ , ≤ γ d τ ( T ) + 2 Cτ k − k A k [ k +1] .
8y (11) and applying Lemma 6, we get for some constant C k depending only on kγ d [ k ] ( T ) ≤ C k (cid:0) γ d τ ( T ) + τ k − k A k [ k +1] + τ − δ d [ k ] ( T ) (cid:1) , ≤ C k (cid:0) γ d τ ( T ) + τ k − k A k [ k +1] + τ − k A k { }···{ k +1 } (cid:1) . (13)Indeed δ d [ k ] ( T ) = 2 sup x ∈ T kh A, x ik { k +1 } = 2 k A k { }···{ k +1 } . We have not used the induction assumption yet. Let I = { i , . . . , i p } be a subset of [ k ],different from ∅ and [ k ]. For j ∈ [ p ] let F ′ j = F i j and let F ′ p +1 = F [ k +1] \ I . Since p < k we canapply inductively Theorem 8 to the tensor A seen as an element of F ′ [ p +1] . For all τ ∈ (0 , γ d I ( T ) ≤ β ′ p X Q τ p − card Q k A k Q , (14)where the sum runs over all partitions Q of [ k + 1] such that the partition { i } , . . . , { i p } , [ k +1] \ I is finer than Q . Again, the inequality is still true if we take the sum over all partitionsof [ k + 1] instead. According to Lemma 7 and since γ is clearly homogeneous, we have γ d τ ( T ) ≤ ρ ′ √ N X ∅ ( I ( [ k ] τ k − card I γ d I ( T )where N is the number of subsets of [ k ] which are different from ∅ and [ k ], namely 2 k −
2. By(14) we get γ d τ ( T ) ≤ D k X P τ k − card P k A k P , for some D k depending only on k . This, together with (13), concludes the proof of Theorem 8.In the last section we prove Lemma 9, this is essentially Lata la’s proof. Let x = ( x , . . . , x k ) ∈ F × · · · × F k , let | x i | be the norm of x i in F i . Let X , . . . , X k beindependant standard Gaussian vectors on F , . . . , F k , respectively. Lemma 10.
For all semi-norm k·k on F [ k ] , we have P (cid:16) k X [ k ] − x [ k ] k ≤ E X ∅ ( I ⊂ [ k ] card I k X I ⊗ x [ k ] \ I k (cid:17) ≥ − k e − P ki =1 | x i | . Proof.
Let us start with an elementary remark. Let x ∈ R n , let K be a symmetric subset of R n , and γ n be the standard Gaussian measure on R n . Then γ n ( x + K ) ≥ γ n ( K )e − | x | . (15)Indeed, the symmetry of K the convexity of the exponential function imply that Z x + K e − | z | dz = Z K (e − | x + y | + e − | x − y | ) dy ≥ Z K e − ( | x | + | y | ) dy k . If k = 1, applying (15) to K = { y ∈ F , k y k ≤ k X k} and x = x , we getP (cid:0) k X − x k ≤ k X k (cid:1) ≥ e − | x | P (cid:0) k X k ≤ k X k (cid:1) . Besides, by Markov we have P (cid:0) k X k ≥ k X k (cid:1) ≤ ≤ , hence the result for k = 1.Let k ≥ k −
1. Let S = X ∅ ( I ⊂ [ k − card I k X I ⊗ x [ k − \ I ⊗ X k k T = X ∅ ( I ⊂ [ k − card I k X I ⊗ x [ k − \ I ⊗ x k k and let A , B and C be the events A = (cid:8) k x [ k − ⊗ ( X k − x k ) k ≤ k x [ k − ⊗ X k k (cid:9) B = (cid:8) k ( X [ k − − x [ k − ) ⊗ X k k ≤ E( S | X k ) (cid:9) C = (cid:8) E( S | X k ) ≤ S + E T (cid:9) . By the following triangle inequality k X [ k ] − x [ k ] k ≤ k x [ k − ⊗ ( X k − x k ) k + k ( X [ k − − x [ k − ) ⊗ X k k , when A , B and C occur we have k X [ k ] − x [ k ] k ≤ k x [ k − ⊗ X k k + 4 E S + E T = E X ∅ ( I ⊂ [ k ] card I k X I ⊗ x [ k ] \ I k . Assume that X k is deterministic, and apply the induction assumption to the spaces F , . . . , F k − and to the semi-norm k y k = k y ⊗ X k k for all y ∈ F [ k − , thenP( B | X k ) ≥ − k +1 e − P k − i =1 | x i | . Since A and C depend only on X k , this implies thatP( A ∩ B ∩ C ) ≥ P( A ∩ C )2 − k +1 e − P k − i =1 | x i | . So it is enough to prove that P( A ∩ C ) ≥ − e − | x k | . For all y ∈ F k we let k y k = k x [ k − ⊗ y k , k y k = E X ∅ ( I ⊂ [ k − card I k X I ⊗ x [ k − \ I ⊗ y k . So that A = (cid:8) k X k − x k k ≤ k X k k (cid:9) ,C = (cid:8) k X k k ≤ k X k k + k x k k (cid:9) . K = { y ∈ F k , k y k ≤ k X k k } ∩ { y ∈ F k , k y k ≤ k X k k } , then, by the triangle inequality, the event X k ∈ x k + K is included in A ∩ C . Using (15), weget P( A ∩ C ) ≥ P( X k ∈ x k + K ) ≥ e − | x k | P( X k ∈ K ) . Therefore, it is enough to prove that P( X k ∈ K ) ≥ , and this is a simple application ofMarkov again.Let F k +1 be another Euclidean space and let A ∈ F [ k +1] . Recall that for I = { i , . . . , i p } ⊂ [ k + 1], we let F I = F i ⊗ · · · ⊗ F i p and k·k I be the corresponding (Euclidean) norm. Our purpose is to apply the previouslemma to the semi-norm defined by k y k = kh A, y ik { k +1 } , for all y ∈ F [ k ] . Notice that for all x ∈ F × · · · × F k and for all ∅ ( I ( [ k ]E k X I ⊗ x [ k ] \ I k ≤ (cid:0) E k X I ⊗ x [ k ] \ I k (cid:1) / = kh A, x [ k ] \ I ik I ∪{ k +1 } , which, according to the definition (8), is equal to d [ k ] \ I (0 , x ). In the same way, when I = [ k ]E kh A, X [ k ] ik { k +1 } ≤ k A k [ k +1] . We let the reader check that Lemma 10 then implies the following: for all τ ∈ (0 ,
1) and x ∈ T , letting ǫ x = d τ ( x,
0) + τ k k A k [ k +1] , we haveP (cid:0) d [ k ] ( x, τ X ) ≤ ǫ x / (cid:1) ≥ − c k τ − (16)for some constant c k depending only on k .Lemma 9 follows easily from this observation. Indeed let S ⊂ T , since S and its translates havethe same entropy numbers, we can assume that 0 ∈ S . Then ǫ x ≤ ǫ := δ d τ ( S ) + τ k k A k [ k +1] for all x ∈ S . Let S ′ be a subset of S satisfying(i) For all x, y ∈ S ′ , d [ k ] ( x, y ) ≥ ǫ .(ii) The set S ′ is maximal (for the inclusion) with this property.By maximality S ′ is an ǫ -net of S , so N ( S, d [ k ] , ǫ ) ≤ card S ′ . On the other hand, by (i) theballs (for d [ k ] ) of radius ǫ/ S ′ do not intersect. This, togetherwith (16), implies that2 − c k τ − card S ′ ≤ X x ∈ S ′ P (cid:0) d [ k ] ( x, τ X ) ≤ ǫ/ (cid:1) ≤ , hence the result. 11 eferences [1] Fernique, X.: Fonctions al´eatoires gaussiennes, vecteurs al´eatoires gaussiens. Centrede Recherches Math´ematiques (1997).[2] Hanson, D.L., Wright, F.T. : A bound on tail probabilities for quadratic forms ofindependant random variables. Ann. Math. Satist., , 1079–1083 (1971).[3] Lata la,R.: Estimates of moments and tails of gaussian chaoses. Ann. Prob., (6),2315–2331 (2006).[4] Ledoux, M.: The concentration of measure phenomenon. American MathematicalSociety (2001).[5] de la Pe˜na V.M., Montgomery-Smith, S.: Bounds for the tail probabilities of U -statistics and quadratic forms. Bull. Amer. Math. Soc., , 223–227 (1994).[6] Talagrand, M.: Regularity of gaussian process. Acta Math.,159