Lower bounds for variances of Poisson functionals

TL;DR

This paper establishes a new lower bound for the variance of Poisson functionals using Malliavin calculus, aiding in the derivation of central limit theorems for stochastic geometry applications.

Contribution

It introduces a novel variance lower bound for Poisson functionals based on the difference operator, with applications to spatial random graphs and Poisson shot noise processes.

Findings

Provides a lower variance bound for Poisson functionals

Demonstrates positive definiteness of asymptotic covariance matrices

Supports multivariate normal approximation results

Abstract

Lower bounds for variances are often needed to derive central limit theorems. In this paper, we establish a lower bound for the variance of Poisson functionals that uses the difference operator of Malliavin calculus. Poisson functionals, i.e. random variables that depend on a Poisson process, are frequently studied in stochastic geometry. We apply our lower variance bound to statistics of spatial random graphs, the $L^p$ surface area of random polytopes and the volume of excursion sets of Poisson shot noise processes. Thereby we do not only bound variances from below but also show positive definiteness of asymptotic covariance matrices and provide associated results on the multivariate normal approximation.