# On Bernstein Type Inequalities for Stochastic Integrals of Multivariate Point Processes

## TL;DR

This paper establishes Bernstein-type concentration inequalities for stochastic integrals of multivariate point processes, providing new bounds and convergence rates for related martingales and estimators.

## Contribution

It introduces novel Bernstein inequalities for multivariate point process integrals and applies them to improve convergence rate results for nonparametric MLEs.

## Key findings

- Derived a Bernstein-type concentration inequality using Doléans-Dade exponential formula.
- Established a uniform exponential inequality via generic chaining.
- Provided an improved convergence rate for nonparametric maximum likelihood estimators.

## 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 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.