On Bernstein type inequalities for stochastic integrals of multivariate point processes

Abstract

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 Doléans-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.

On De la Peña Type Inequalities for Point Processes

“…is a local martingale. Following the similar arguments in Wang Lin and Su [19], we have e Xt /E(S(λ)) t t≥0 is a local martingale. In fact, set H = e X , G = E(S(λ)), A = S(λ) and f (h, g) = h g .…”