On De la Pe\~{n}a Type Inequalities for Point Processes

TL;DR

This paper extends de la Peña's exponential inequalities to stochastic integrals of multivariate point processes, providing new tools for analyzing the concentration of such processes with applications to coalescent models.

Contribution

It introduces de la Peña type inequalities for multivariate point processes, expanding the scope of concentration inequalities in stochastic process theory.

Findings

Derived exponential inequalities for stochastic integrals of multivariate point processes

Applied inequalities to block counting processes in Lambda-coalescents

Enhanced understanding of concentration properties in complex stochastic models

Abstract

There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Pe\~{n}a \cite{d} establishes a nice exponential inequality for discrete time locally square integrable martingale . In this paper, we obtain de la Pe\~{n}a's inequalities for stochastic integral of multivariate point processes. The proof is primarily based on Dol\'{e}ans-Dade exponential formula and the optional stopping theorem. As application, we obtain an exponential inequality for block counting process in $\Lambda-$coalescents.