Exponential inequalities for the supremum of some counting processes and their square martingales

Ronan Le Guével

doi:10.5802/crmath.206

Abstract

We establish exponential inequalities for the supremum of martingales and square martingales obtained from counting processes, as well as for the oscillation modulus of these processes. Our inequalities, that play a decisive role in the control of errors in statistical procedures, apply to general non-explosive counting processes including Poisson, Hawkes and Cox models. Some applications for $U$-statistics are discussed.

Highlights

Counting processes naturally arise in a lot of applied fields and the understanding of their evolution is the object of a lot of modelling problems
Exponential inequalities for the distribution of random variables have been of interest for many years, and they are still a very active research area for various types of processes, like sums of i.i.d. random variables, empirical processes, U -statistics, Poisson processes, martingales and self-normalised martingales, with discrete or continuous time
For discrete time processes with i.i.d. random variables, exponential inequalities have been obtained for the empirical process or for U -statistics of order two in [1, 7, 11, 12, 16, 18, 19, 25] or [10] to cite a few

Summary

Introduction

Counting processes naturally arise in a lot of applied fields and the understanding of their evolution is the object of a lot of modelling problems. Our aim is to provide exponential inequalities with explicit constants for general counting processes, for their associated local martingales, for their local square martingales and for their oscillation modulus. The Poisson process is seen as a point process (Ti )i ≥1 on the real line, allowing to use the inequalities obtained for U -statistics of i.i.d. random variables like Rosenthal’s inequality and Talagrand’s inequality, after conditioning by the total random number of point. This approach is no longer valid when we consider more general counting processes than the Poisson process.

Notations

Martingale properties

Exponential inequalities

U-statistics of order two

Oscillation modulus control

Proofs