A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

Abstract

In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdös-Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random $\kappa$-complex and infinite weighted 2-runs are treated.

References in corpus (7)

• Central limit theorems for sequences of multiple stochastic integrals
• Stochastic analysis of Bernoulli processes
• Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs
• Fourth moment theorems on the Poisson space in any dimension
• Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals
• A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph
• A Peccati-Tudor type theorem for Rademacher chaoses