On Bernstein type inequalities for stochastic integrals of multivariate point processes

In this paper, we first obtain a Bernstein type of concentration inequality for stochastic integrals of multivariate point processes under some conditions through the Doléans-Dade exponential formula, and then derive a uniform exponential inequality using a generic chaining argument. As a direct consequence, we obtain an upper bound for a sequence of discrete time martingales indexed by a class of functionals. Finally, we apply the uniform exponential bound to nonparametric maximum likelihood estimators and provide a rate of convergence in terms of Hellinger distance, which is an improvement of earlier work of van de Geer (1995).