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

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For instance, one deduces immediately from the previous expression that d dx H n (x) = nH n 1 (x) , n 1, (5.9) H n+1 (x) = xH n (x) nH n 1 (x) , n 1. (5.10) The next result uses (5.5) and (5.10) in order to establish an explicit relation between multiple stochastic integrals and Hermite polynomials. Proposition 5.1 Let h 2 L 2 () be such that khk L 2 () = 1, and, for n 2, de…ne Then, h n (z 1 ; ::; z n ) = h (z 1 ) h (z n ) , (z 1 ; :::; z n ) 2 Z n . I n h n = H n (G (h)) = H n (I 1 (h)) : (5.11) Proof. Of course, H 1 (I 1 (h)) = I 1 (h). By the multiplication formula (5.5), one has therefore that, for n 2, I n h n I 1 (h) = I n+1 h n+1 + nI n 1 h n 1 ; and the conclusion is obtained from (5.10), and by recursion on n. Remark. By using the relation E [I n (h n ) I n (g n )] = n! hh n ; g n i L 2 ( n ) = n! hh; gin L 2 () , we infer from (5.11) that, for every jointly Gaussian random variables (U; V ) with zero mean and unitary variance, 0 if m 6= n E [H n (U) H m (V )] = n!E [UV ] n if m = n: 5.3 Chaotic decompositions By combining (5.8) and (5.11), one obtains the following fundamental decomposition of the square-integrable functionals of G. Theorem 5.2 (Chaotic decomposition) For every F 2 L 2 ( (G) ; P) (that is, F is a squareintegrable functional of G), there exists a unique sequence ff n : n 1g, with f n 2 L 2 s ( n ), such that 1X F = E [F ] + I n (f n ) , (5.12) n=1 where the series converges in L 2 (P). Proof. Fix h 2 L 2 () such that khk L 2 () = 1, as well as t 2 R. By using (5.8) and (5.11), one obtains that t 2 1X t n 1X exp tG (h) = 2 n! H t n n (G (h)) = 1 + n! I n h n : (5.13) n=0 18 n=1
h i t Since E exp tG (h) 2 2 = 1, one deduces that (5.12) holds for every random variable of the t form F = exp tG (h) 2 2 , with f n = tn n! hn . The conclusion is obtained by observing that the linear combinations of random variables of this type are dense in L 2 ( (G) ; P) : Remarks. (1) Proposition 4.2, together with (5.12), implies that E F 2 = E [F ] 2 + 1X n! kf n k 2 L 2 ( n ) . (5.14) n=1 (2) By using the notation (4.6), (4.13) and (4.14), one can reformulate the statement of Theorem 5.2 as follows: L 2 ( (G) ; P) = 1M C n (G) , n=0 where “ ”indicates an in…nite orthogonal sum in L 2 (P). (3) By inspection of the proof of Theorem 5.2, we deduce that the linear combinations of random variables of the type I n (h n ), with n 1 and khk L 2 () = 1, are dense in L2 ( (G) ; P). This implies in particular that the random variables I n (h n ) generate the nth Wiener chaos C n (G). (4) The …rst proof of (5.12) dates back to Wiener [99]. See also McKean [50], Nualart and Schoutens [68] and Stroock [91]. See e.g. [19], [35], [39], [43] and [72] for further references and results on chaotic decompositions. 6 Isonormal Gaussian processes In this section we brie‡y show how to generalize the previous results to the case of an isonormal Gaussian process. These objects have been introduced by Dudley in [22], and are a natural generalization of the Gaussian measures introduced above. In particular, the concept of an isonormal Gaussian process can be very useful in the study of fractional …elds. See e.g. Pipiras and Taqqu [76, 77, 78], or the second edition of Nualart’s book [65]. For a general approach to Gaussian analysis by means of Hilbert space techniques, and for further details on the subjects discussed in this section, the reader is referred to Janson [35]. 6.1 General de…nitions and examples Let H be a real separable Hilbert space with inner product h; i H . In what follows, we will denote by X = X (H) = fX (h) : h 2 Hg an isonormal Gaussian process over H. This means that X is a centered real-valued Gaussian family, indexed by the elements of H and such that E X (h) X h 0 = h; h 0 H , 8h; h0 2 H: (6.1) 19
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: By an application of the Cauchy-Sch
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 and 48: To prove the Wasserstein estimate (
Page 49 and 50: 2. 4 (F ) = E F 4 q 1 X 3 = 3q p=
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

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