On De la Pe\~{n}a 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 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 $\mathrm{\Lambda }-$coalescents.