Large deviations for fractional Poisson processes

TL;DR

This paper establishes large deviation principles for two types of fractional Poisson processes, analyzing their implications for ruin probabilities in insurance models and exploring their properties with Mittag Leffler functions.

Contribution

It introduces large deviation results for fractional Poisson processes, including renewal and weighted versions, and applies these to insurance risk models.

Findings

Large deviation principles are proven for fractional Poisson renewal processes.

Explicit estimates for ruin probabilities in insurance models with fractional Poisson claims.

Analysis of weighted Poisson distributed variables in the context of fractional processes.

Abstract

We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d. light tail claim sizes, and a fractional Poisson claim number process. We conclude with the alternative version where all the random variables are weighted Poisson distributed. Keywords: Mittag Leffler function; renewal process; random time cha