A Fractional Generalization of the Poisson Processes and Some of its Properties

TL;DR

This paper introduces a fractional generalization of Poisson processes using fractional derivatives, explores its properties, and discusses its applications to time series and related stochastic models.

Contribution

It presents a novel fractional Poisson process model based on Mittag-Leffler distributions and analyzes its properties and relevance to time series analysis.

Findings

Derived a fractional Poisson renewal process model.

Analyzed the process's properties and sample paths.

Introduced the q-Mittag-Leffler process.

Abstract

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order $\alpha ~(0 < \alpha \leq 1)$. A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors and sample path properties are studied. Finally we introduce the $q$-Mittag-Leffler process associated with the $q$-Mittag-Leffler distribution.