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

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.