Skellam Type Processes of Order k and Beyond.

Nikolai Leonenko,Arun Kumar,Neha Gupta

doi:10.3390/e22111193

Nikolai Leonenko, Arun Kumar + Show 1 more

Open Access

https://doi.org/10.3390/e22111193

Copy DOI

Abstract

In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Skellam process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, Lévy measures, governing difference-differential equations of the introduced processes. Our results generalize the Skellam process and running average of Poisson process in several directions.

Highlights

The Skellam distribution is obtained by taking the difference between two independent Poisson distributed random variables, which was introduced for the case of different intensities λ1, λ2 by and for equal means in [2]
For large values of λ1 + λ2, the distribution can be approximated by the normal distribution and if λ2 is very close to 0, the distribution tends to a Poisson distribution with intensity λ1
The Skellam process is an integer valued Lévy process and it can be obtained by taking the difference of two independent Poisson processes

Summary

Introduction

The Skellam distribution is obtained by taking the difference between two independent Poisson distributed random variables, which was introduced for the case of different intensities λ1 , λ2 by (see [1]) and for equal means in [2]. The Skellam process of order k can be used to model the difference between the number of points scored by two competing teams in a basketball match where k = 3. It is shown that the Mittag–Leffler distribution is a better model than the exponential distribution for the inter-arrival times between the up and down jumps. We hope that a more flexible class of processes like time-changed Skellam process of order k (see Section 6) and the introduced tempered space-fractional Skellam process (see Section 7) would be better model for arrivals of jumps.

Preliminaries

Skellam Process

Subordinators

The Space-Fractional Poisson Process

Tempered Space-Fractional Poisson Process

Running Average of PPoK

Running Average of SPoK

Time-Changed Skellam Process of Order k

The Space-Fractional Skellam Process

Simulation of SFSP and TSFSP