Tempered space-time fractional negative binomial process

TL;DR

This paper introduces a new tempered space-time fractional negative binomial process (TSTFNBP), analyzing its distribution, dependence, overdispersion, and first-passage times, with implications for modeling overdispersed count data.

Contribution

It defines the TSTFNBP by subordinating fractional Poisson with a tempered Mittag-Leffler Lévy process and explores its properties and applications.

Findings

TSTFNBP exhibits overdispersion.

It has long-range dependence.

Asymptotic fractional moments are derived.

Abstract

In this paper, we define a tempered space-time fractional negative binomial process (TSTFNBP) by subordinating the fractional Poisson process with an independent tempered Mittag-Leffler L\'{e}vy subordinator. We study its distributional properties and its connection to partial differential equations. We derive the asymptotic behavior of its fractional order moments and long-range dependence property. It is shown that the TSTFNBP exhibits overdispersion. We also obtain some results related to the first-passage time.