Dimension free and infinite variance tail estimates on Poisson space

TL;DR

This paper develops concentration inequalities on Poisson space applicable to functionals with finite or infinite variance, providing dimension-free tail bounds and exponential integrability results, with applications to stochastic processes.

Contribution

It introduces novel tail estimates on Poisson space that handle both finite and infinite variance cases, including stable tail bounds for infinitely divisible variables.

Findings

Dimension-free tail estimates for Poisson functionals

Exponential integrability results for Euclidean norms

Tail bounds for stable and infinite variance cases

Abstract

Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including L\'evy's stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented.