Second order Poincaré inequalities and CLTs on Wiener space

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Niveau: Supérieur, Licence, Bac+2
Second order Poincaré inequalities and CLTs on Wiener space by Ivan Nourdin ? , Giovanni Peccati † and Gesine Reinert ‡ Université Paris VI, Université Paris Ouest and Oxford University Abstract: We prove infinite-dimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Stein's method and Malliavin calculus. We provide two applications: (i) to a new second order characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields. Key words: central limit theorems; isonormal Gaussian processes; linear functionals; multi- ple integrals; second order Poincaré inequalities; Stein's method; Wiener chaos 2000 Mathematics Subject Classification: 60F05; 60G15; 60H07 1 Introduction Let N ? N (0, 1) be a standard Gaussian random variable. In its most basic formulation, the Gaussian Poincaré inequality states that, for every di?erentiable function f : R? R, Varf(N) 6 Ef ?(N)2, (1.1) with equality if and only if f is a?ne. The estimate (1.1) is a fundamental tool of stochastic analysis: it implies that, if the random variable f ?(N) has a small L2(?) norm, then f(N) has necessarily small fluctuations.

  • dimensional poincaré

  • isonormal gaussian process

  • standard gaussian

  • malliavin

  • multi-dimensional

  • operator ?

  • gaussian subordinated

  • random variable

  • poincaré inequalities


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