Multivariate central limit theorems for Rademacher functionals with applications

TL;DR

This paper develops multivariate central limit theorems for Rademacher functionals using discrete Malliavin calculus and applies these results to random graphs and cubical complexes, providing quantitative bounds.

Contribution

It introduces a discrete multivariate second-order Poincaré inequality and applies it to complex combinatorial and geometric structures.

Findings

Normal approximation for subgraph counts in Erdős-Rényi graphs

Quantitative CLT for vectors of vertex degrees

CLT for intrinsic volumes of random cubical complexes

Abstract

Quantitative multivariate central limit theorems for general functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal approximation. In particular, a discrete multivariate second-order Poincar\'e inequality is developed. As a first application, the normal approximation of vectors of subgraph counting statistics in the Erd\H{o}s-R\'enyi random graph is considered. In this context, we further specialize to the normal approximation of vectors of vertex degrees. In a second application we prove a quantitative multivariate central limit theorem for vectors of intrinsic volumes induced by random cubical complexes.