Non-central moderate deviations for compound fractional Poisson processes

TL;DR

This paper investigates non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps, filling a gap between large deviations and weak convergence to non-Gaussian limits.

Contribution

It introduces the concept of non-central moderate deviations in the context of compound fractional Poisson processes, extending the understanding of deviation principles beyond Gaussian limits.

Findings

Characterization of non-central moderate deviations for the processes

Identification of non-Gaussian limiting distributions

Extension of deviation principles to fractional Poisson processes

Abstract

The term "moderate deviations" is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about "non-central moderate deviations" when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.