On Bernstein Type Inequalities for Stochastic Integrals of Multivariate Point Processes

We consider the stochastic integrals of multivariate point processes and study their concentration phenomena. In particular, we obtain a Bernstein type of concentration inequality through Dol'eans-Dade exponential formula and a uniform exponential inequality using a generic chaining argument. As applications, we obtain a upper bound for a sequence of discrete time martingales indexed by a class of functionals, and so derive the rate of convergence for nonparametric maximum likelihood estimators, which is an improvement of earlier work of van de Geer.