On De la Pena Type Inequalities for Point Processes

There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Pena established a nice exponential inequality for a discrete time locally square integrable martingale. In this paper, we obtain de la Pena's inequalities for a stochastic integral of multivariate point processes. The proof is primarily based on Doleans-Dade exponential formula and the optional stopping theorem. As an application, we obtain an exponential inequality for block counting process in A-coalescent.