Exponential inequalities and laws of the iterated logarithm for multiple Poisson--Wiener integrals and Poisson $U$-statistics

TL;DR

This paper establishes tail and moment inequalities for Poisson stochastic integrals and $U$-statistics, demonstrating the Law of the Iterated Logarithm for these processes as the Poisson intensity grows, with applications to various Poisson functionals.

Contribution

It introduces new tail and moment inequalities for Poisson integrals and $U$-statistics, enabling the proof of the Law of the Iterated Logarithm in this context.

Findings

Law of the Iterated Logarithm proven for Poisson processes with increasing intensity

Improved concentration of measure results for classical Poisson functionals

Applications to geometric graphs, Poisson $k$-flats, and Lévy processes

Abstract

We prove tail and moment inequalities for multiple stochastic integrals on the Poisson space and for Poisson $U$-statistics. We use them to demonstrate the Law of the Iterated Logarithm for these processes when the intensity of the Poisson process tends to infinity, with normalization depending on the degree of the multiple stochastic integral or degeneracy of the kernel defining the $U$-statistic. We apply our results to several classical functionals of Poisson point processes, obtaining improvements or complements of known concentration of measure results as well as new laws of the iterated logarithm. Examples include subgraph counts and power length functionals of geometric random graphs, intersections of Poisson $k$-flats, quadratic functionals of the Ornstein--Uhlenbeck L\'evy process and $U$-statistics of marked processes. Keywords: Poisson point process, $U$-statistics, multiple stochastic integrals, concentration of measure, The Law of the Iterated Logarithm