A simplified second-order Gaussian Poincar\'e inequality in discrete setting with applications

TL;DR

This paper introduces a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals of Rademacher variables, with applications to random graphs, hypercubes, and complexes.

Contribution

It develops a new bound for the Kolmogorov distance using the discrete Malliavin-Stein method, advancing normal approximation techniques in discrete probability.

Findings

Bound for Kolmogorov distance established

Applications to Erdős-Rényi graphs and hypercube percolation

Analysis of random complexes and weighted 2-runs

Abstract

In this paper, a simplified second-order Gaussian Poincar\'e 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 Erd\"os-R\'enyi 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.