Stein's method, Malliavin calculus and infinite-dimensional Gaussian

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If q = 2, then there exists a sequence f i : i 1g of i.i.d. centered standard Gaussian random variables such that 1X I 2 (f) = i 2 i 1 , (10.1) i=1 where the series converges in L 2 (P), and f i : i 1g is the sequence of eigenvalues of the Hilbert-Schmidt operator (from H into H) given by h 7! f 1 h, where 1 indicates a contraction of order 1. In particular, I 2 (f) admits some …nite exponential moment, and the law of I 2 (f) is determined by its moments. If q 3, the law of I q (f) may not be determined by its moments. See Slud [87]. For every q 2, the random variable I q (f) cannot be Gaussian. See [35, Chapter VI]. For q 3, and except for trivial cases, there does not exist a general explicit formula for the characteristic function of I q (f). Note that in the next section we will focus on the total variation distance d T V . However, it will be clear later on that (thanks to Theorem 9.1) all the results extend without di¢ culties to the Fortet-Mourier, Wasserstein or Kolmogorov distances. 10.1 CLTs in dimension one Let N N (0; 1). Fix q 2 and consider an element of the qth Wiener chaos of X with the form F = I q (f), where the kernel f is in H q . Remark. Since E [F ] = 0, the fourth cumulant of F is given by 4 (F ) = E F 4 3E F 2 2 . (10.2) Observe also that E N 4 = 3. We have that E h 1 q kDF k2 H i = E F 2 = q! kfk 2 Hq, and also, by (9.4), d T V (F; N) 2 1 q! kfk 2 1 H + 2 sVar q q kDF k2 H . (10.3) The following result, which is partially based on the moment formula (7.35), shows that (10.3) yields indeed an important simpli…cation of the method of moments and cumulants. Proposition 10.1 (See [57]) Let the above notation and assumptions prevail. Then, the following hold. 1. 1 Var q kDF Xq 1 q 1 4 k2 H = q 2 (p 1)! 2 (2q 2p)! f e p f 2 : (10.4) p 1 H 2(q p) p=1 48
2. 4 (F ) = E F 4 q 1 X 3 = 3q p=1 q p! (p 1)! p 2 q 1 p 1 2 (2q 2p)! f e p f 2 H 2(q p) : (10.5) 3. 4. 0 1 3q 4 (F ) Var 1 q kDF k2 H d T V (N; F ) 2 1 E F 2 + q 1 3q 4 (F ) : (10.6) r q 1 3q 4 (F ) . (10.7) Proof. It su¢ ces to prove the statement when H = L 2 (Z; Z; ), with -…nite and without atoms. In this case, one has that D z F = qI q 1 (f (; z)) and, by the multiplication formula, Xq 1 (D z F ) 2 q 1 2 = q 2 r! I r 2(q 1 r) (f (; z) r f (; z)) : It follows that r=0 1 q kDF k2 L 2 () = 1 q Z Z q 1 (D z F ) 2 (dz) X q 1 2 = q r! I r 2(q 1 r) (f r+1 f) r=0 qX q 1 2 = q (p 1)! I p 1 2(q p) (f p f) , (10.8) p=1 so that (10.4) follows immediately from the isometry and orthogonality properties of multiple Wiener-Itô integrals. To prove (10.5), we start by observing that, thanks to (7.35) in the case n = 2, E F 4 = 3 F 2 1 q kDF k2 L 2 () : (10.9) Now, by virtue of the multiplication formula, F 2 = qX p=0 q 2 p! I p 2(q p) (f p f) , and, by plugging (10.8) into (10.9), we obtain E F 4 = 3q qX p=1 q 2 q 1 2 p! (p 1)! (2q 2p)! f e p f 2 , p p 1 H 2(q p) 49
Page 1 and 2: Stein’s method, Malliavin calculu
Page 3 and 4: approximation (see e.g. [14], [80],
Page 5 and 6: As discussed in [37], and later in
Page 7 and 8: obtained by juxtaposing n copies of
Page 9 and 10: D4 The application of the DDS Theor
Page 11 and 12: Notation. For the rest of the paper
Page 13 and 14: matrix (i; j) = hf i ; f j i L 2 ()
Page 15 and 16: 1. For every n, the application f 7
Page 17 and 18: By an application of the Cauchy-Sch
Page 19 and 20: h i t Since E exp tG (h) 2 2 = 1,
Page 21 and 22: This type of construction is used i
Page 23 and 24: Proposition 6.1 1. For every q 1,
Page 25 and 26: Cp 1 (R n ) is the class of in…ni
Page 27 and 28: We also set D 1;2 = 1\ D k;2 (7.9)
Page 29 and 30: Maxima. Suppose that X (h i ), i =
Page 31 and 32: denote by F t the …ltration gener
Page 33 and 34: on the other hand, by using (5.8),
Page 35 and 36: Theorem 7.1 For every F 2 L 2 ( (X)
Page 37 and 38: This implies that the conditional e
Page 39 and 40: Lemma 8.1 (i) Let W be a random var
Page 41 and 42: 8.3 Multi-dimensional Stein’s Lem
Page 43 and 44: On the other hand, by noting h A 1(
Page 45 and 46: where, for every v = (v 1 ; :::; v
Page 47: To prove the Wasserstein estimate (
Page 51 and 52: [21]), quadratic functionals of fra
Page 53 and 54: (i) For every 1 i d, F (n) i conv
Page 55 and 56: 11 Two examples The theory develope
Page 57 and 58: (n) 3 q (n) 4 6 + ( (n) 2 1) 2 !
Page 59 and 60: Theorem 11.1 Fix > 0 and q 2, and
Page 61 and 62: References [1] P. Baldi, G. Kerkyac
Page 63 and 64: [34] K. Itô (1951). Multiple Wiene
Page 65 and 66: [71] G. Peccati and M.S. Taqqu (200

Stein's method, Malliavin calculus and infinite-dimensional Gaussian

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