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

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10.3 A non-central limit theorem (with bounds) We now present (without proofs) two statements concerning the Gamma approximation of multiple integrals of even order q 2. The …rst result, which is taken from [57], provides an explcit representation for the quantities appearing on the RHS of (9.11) and (9.12). Proposition 10.4 (See [57]) Let q 2 be an even integer, and let G = I q (g), where g 2 H q . Then, where E[(2 + 2G hDG; DL 1 Gi H ) 2 ] = E[(2 + 2G q 1 kDGk 2 H )2 ] (10.15) (2 q! kgk 2 H q)2 + +q 2 X 1 c q = (q=2)! q 1 q=2 1 r2f1;:::;q 1g r6=q=2 4 2 = (q=2)! q 1 4 (2q 2r)!(r 1)! 2 kg r gk 2 + r 1 H 2(q r) q q=2 +4q! c 1 q g e q=2 g g 2 H q ; 2 : (10.16) The next statement, which is a main result of [56], contains a “non-central” analogous of Theorem 10.1. Recall the de…nition of the centered Gamma random variables F (), > 0, given in (8.2). Theorem 10.3 (See [56]) Fix > 0, as well as an even integer q 2. De…ne c q as in (10.16). Then, for any sequence ff k g k1 H q verifying lim q!kf kk 2 H = lim E I k!1 n q (f k ) 2 = Var (F ()) = 2; (10.17) k!1 the following six conditions are equivalent: (i) lim k!1 E[I q (f k ) 3 ] = E[F () 3 ] = 8 and lim k!1 E[I q (f k ) 4 ] = E[F () 4 ] = 48 + 12 2 ; (ii) lim k!1 E[I q (f k ) 4 ] 12E[I q (f k ) 3 ] = 12 2 48; (iii) lim k!1 kf k e q=2 f k c q f k k H q = 0 and lim k!1 kf k e p f k k H 2(q p) = 0, for every p = 1; :::; q 1 such that p 6= q=2; (iv) lim k!1 kf k e q=2 f k c q f k k H q = 0 and lim k!1 kf k p f k k H 2(q p) = 0, for every p = 1; :::; q 1 such that p 6= q=2; (v) as k ! 1, kD[I q (f k )]k 2 H 2qI q (f k ) ! 2q in L 2 ; (vi) as k ! 1, the sequence fI q (f k )g k1 converges in distribution to F (). Remark. In [56], Theorem 10.3 is not proved with Stein’s method, but rather by implementing the “di¤erential approach” initiated by Nualart and Ortiz-Latorre in [66]. However, it is not di¢ cult to see that (9.11), (9.12) and (10.15) can be combined in order to deduce an alternate proof of the implications (iv) ) (v) ) (vi). 54
11 Two examples The theory developed in the previous sections (along with its re…nements and generalizations – see Section 12) has been already applied in a variety of frameworks. In particular: In [57], Theorem 9.1 and Proposition 10.1 are applied in order to deduce explicit Berry- Esséen bounds for the so-called Breuer-Major CLT (see [5]), involving Hermite-type transformations of fractional Brownian motion. This analysis is further developed in [4], [58] and [60] The paper [58] contains applications to Toepliz quadratic forms in continuous time –see e.g. [30] and the references therein. In [60] one can also …nd multidimensional generalizations of Chatterjee’s result (9.7). Reference [62] contains an application of (9.3) to the proof of in…nite-dimensional secondorder Poincaré inequalities on Wiener space. In [64], relation (7.30) is exploited in order to provide a new explicit expression for the densities of functionals of isonormal Gaussian processes. In [97], one can …nd applications to tail bounds on Gaussian functionals and polymer models. Remark. Apart from the previous references, the applications to fractional Brownian motion and density estimation are discussed in the lecture notes [53]. In what follows, we shall present two further applications of the previous results. The …rst one (basically taken from [58]) focuses on exploding quadratic functionals of a Brownian sheet – thus completing the discussion contained in Section 2. The second one involves Hermite transformations of multiparameter Ornstein-Uhlenebck Gaussian processes, and is new (albeit it is inspired by the last section of [70]). 11.1 Exploding Quadratic functionals of a Brownian sheet 11.1.1 Statement of the problem Let d 1, and let W = nW (t 1 ; :::; t d ) : (t 1 ; :::; t d ) 2 [0; 1] do be a standard Brownian sheet on [0; 1] d . Recall that this means that W is a continuous centered Gaussian process with a covariance function given by E [W (t 1 ; :::; t d ) W (s 1 ; :::; s d )] = dY (t j ^ s j ) : j=1 55
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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 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: (i) For every 1 i d, F (n) i conv
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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Stein's method, Malliavin calculus and infinite-dimensional Gaussian

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