Noncentral moderate deviations for fractional Skellam processes

TL;DR

This paper investigates noncentral moderate deviations for fractional Skellam processes, revealing their convergence behaviors and establishing inequalities between rate functions, thus extending the understanding of large deviation principles beyond Gaussian limits.

Contribution

It provides new noncentral moderate deviation results for fractional Skellam processes and compares convergence rates with type 2 processes using rate function inequalities.

Findings

Noncentral moderate deviations are established for fractional Skellam processes.

Convergence to zero is faster for fractional Skellam process of type 2.

Rate function inequalities demonstrate improved convergence rates.

Abstract

The term \emph{moderate deviations} is often used in the literature to mean a class of large deviation principles that, in some sense, fills 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 \emph{noncentral moderate deviations} when the weak convergence is towards a non-Gaussian distribution. In this paper we present noncentral moderate deviation results for two fractional Skellam processes in the literature (see Kerss, Leonenko and Sikorskii, 2014). We also establish that, for the fractional Skellam process of type 2 (for which we can refer the recent results for compound fractional Poisson processes in Beghin and Macci (2022)), the convergences to zero are usually faster because we can prove suitable inequalities between rate functions.