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

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[53] I. Nourdin (2008). Integration by parts on Wiener space and applications. Lecture notes available at: http: // www.proba.jussieu.fr/ ~nourdin/ibp-wienerspace.pdf [54] I. Nourdin, D. Nualart and C.A. Tudor (2007). Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Preprint. [55] I. Nourdin and G. Peccati (2008). Weighted power variations of iterated Brownian motion. The Electronic Journal of Probability. 13, n. 43, 1229-1256 (Electronic). [56] I. Nourdin and G. Peccati (2008). Non-central convergence of multiple integrals. Ann. Probab., to appear. [57] I. Nourdin and G. Peccati (2008). Stein’s method on Wiener chaos. Probab. Theory Rel. Fields, to appear. [58] I. Nourdin and G. Peccati (2008). Stein’s method and exact Berry-Essén bounds for functionals of Gaussian …elds. Preprint. [59] I. Nourdin and G. Peccati (2009). Limit theorems using Stein’s method and Malliavin calculus. In preparation. [60] I. Nourdin, G. Peccati and A. Réveillac (2008). Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Statist., to appear. [61] I. Nourdin, G. Peccati and G. Reinert (2008). Stein’s method and stochastic analysis of Rademacher functionals. Preprint. [62] I. Nourdin, G. Peccati and G. Reinert (2008). Second order Poincaré inequalities and CLTs on Wiener space. J. Func. Anal., to appear. [63] I. Nourdin and A. Réveillac (2008). Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H = 1=4. Preprint. [64] I. Nourdin and F.G. Viens (2008). Density estimates and concentration inequalities with Malliavin calculus. Preprint. [65] D. Nualart (2006). The Malliavin calculus and related topics. Springer-Verlag, Berlin, 2nd edition. [66] D. Nualart and S. Ortiz-Latorre (2008). Central limit theorem for multiple stochastic integrals and Malliavin calculus. Stoch. Proc. Appl. 118 (4), 614–628. [67] D. Nualart and G. Peccati (2005). Central limit theorems for sequence of multiple stochastic integrals. Ann. Probab. 33 (1), 177–193. [68] D. Nualart and W. Schoutens (2000). Chaotic and predictable representation for Lévy processes. Stoch. Proc. Appl. 90, 109-122. [69] G. Peccati (2007). Gaussian approximations of multiple integrals. Electronic Communications in Probability 12, 350-364 (electronic). [70] G. Peccati, J.-L. Solé, M.S. Taqqu and F. Utzet (2008). Stein’s method and normal approximation of Poisson functionals. Preprint. 64
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Stein’s method, Malliavin calculu
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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 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: [34] K. Itô (1951). Multiple Wiene
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