Second order Poincaré inequalities and CLTs on Wiener space

By Ivan Nourdin, Giovanni Peccati, Gesine Reinert. 2008

90 citations recorded by the source corpus.

We prove infinite-dimensional second order Poincare 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.

Subfields: Applied Mathematics, Finance, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/90a225a75fc3756d2115a4c8baf35ed4cd377c7a

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Citations in the corpus

Stein's method for comparison of univariate distributions (2014, background): “There is an important connection between such constants and Poincaré / variance bounds or spectral gaps, as already noted for example in [23, 56, 54, 61, 72].”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “This class of second order results has been further generalized in [33] to the framework of functionals F of an infinite-dimensional isonormal Gaussian process X over a Hilbert space H, in which case the estimate (1.”
Cumulants on the Wiener Space (2009, methodology): “…results for multivariate normal approximations; [1] focuses on further developments in the 1 multivariate case, in relation with quasi-sure analysis; [16] deals with inﬁnite-dimensional second order Poincar´e inequalities; in [19], one can ﬁnd new explicit expressions for the density of…”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “In the recent series of papers [12, 13, 16], it has been shown that one can effectively combine Malliavin calculus with Stein’s method, in order to evaluate the distance between the law of an element of a fixed Wiener chaos, say, F , and a standard Gaussian distribution.”
A short survey of Stein's method (2014, methodology): “n [34], using a characterization of the normal distribution by Borovkov and Utev [15]. Second orderPoincar´einequalitieshavebeen useful in severalsubsequentworks, e.g. in Nourdin, Peccati and Reinert [62], Nolen [59], etc. Indeed, it may be said that the whole thriving area of Stein’s method in Mall”
Stein couplings for normal approximation (2010, unknown): “In Chatterjee (2009) and Nourdin, Peccati, and Reinert (2009) a connection with Poincaré inequalities was made, where a bound on VarEWT1 is called a Poincaré inequality of second order.”