An Improved Second-Order Poincaré Inequality for Functionals of Gaussian Fields

Abstract

We present an improved version of the second order Gaussian Poincaré inequality, firstly introduced in Chatterjee (2009) and Nourdin, Peccati and Reinert (2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative CLTs for non-linear functionals of stationary Gaussian fields related to the Breuer-Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.

The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications

“…Theorem 1.7 (Theorem 2.1 in [42]) Let F ∈ D 2,4 with mean zero and variance σ 2 > 0 and let Z ∼ N (0, σ 2 ). Suppose H = L 2 (A, ν) with ν a diffusive measure on some measurable space A. Then,…”

Malliavin–Stein method: a survey of some recent developments

“…controls the variance of a random variable F by means of integrated moments of the add-one cost (see [51,Section 18.3]), the integrated moments of second-order addone-cost D + x D + y F := D 2 z,y F controll the discrepancy between the distribution of F and that of a Gaussian random variable -a phenomenon already observed in the Gaussian setting [21,70,96], where gradients typically replace add-one-cost operators.…”

“…As formally discussed in the sections to follow, the basic idea of the approach initiated in [64] is that, in order to assess the discrepancy between some target law (Normal or Gamma, for instance), and the distribution of a nonlinear functional of a Gaussian field, one can fruitfully apply infinitedimensional integration by parts formulae from the Malliavin calculus of variations [57,66,77,78] to the general bounds associated with the so-called Stein's method for probabilistic approximations [66,23]. In particular, the Malliavin-Stein approach captures and amplifies the essence of [21], where Stein's method was combined with finite-dimensional integration by parts formulae for Gaussian vectors, in order to deduce second order Poincaré inequalities -as applied to random matrix models with Gaussian-subordinated entries (see also [70,96]).…”

Malliavin-Stein Method: a Survey of Recent Developments

“…As formally discussed in the sections to follow, the basic idea of the approach initiated in [NP09b] is that, in order to assess the discrepancy between some target law (Normal or Gamma, for instance), and the distribution of a non-linear functional of a Gaussian field, one can fruitfully apply infinite-dimensional integration by parts formulae from the Malliavin calculus of variations [M97,NP12,Nua06,Nua09] to the general bounds associated with the so-called Stein's method for probabilistic approximations [NP12,CGS10]. In particular, the Malliavin-Stein approach captures and amplifies the essence of [C09], where Stein's method was combined with finite-dimensional integration by parts formulae for Gaussian vectors, in order to deduce second order Poincaré inequalities -as applied to random matrix models with Gaussian-subordinate entries (see also [Vid17]).…”