Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes

TL;DR

This paper provides explicit bounds and deviation principles for Poisson shot-noise processes and generalized Hawkes processes, enhancing understanding of their normal approximation and concentration behaviors.

Contribution

It introduces a new class of generalized compound Hawkes processes and derives explicit bounds, moderate deviation principles, and concentration inequalities for them.

Findings

Explicit Wasserstein and Kolmogorov bounds for Poisson chaos variables.

Moderate deviation principles and concentration inequalities established.

Application to generalized compound Hawkes processes extends existing literature.

Abstract

In this article, we give explicit bounds on the Wasserstein and the Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard Normal distribution, using the results proved by Last, Peccati and Schulte. Relying on the theory developed in the work of Saulis and Statulevicius and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities and Normal approximation bounds with Cram\'er correction terms for the same variables. The aforementioned results are then applied to Poisson shot-noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results availale in the literature regarding the Normal approximation and moderate deviations.