Some Compound Fractional Poisson Processes

Abstract

In this paper, we introduce and study fractional versions of the Bell–Touchard process, the Poisson-logarithmic process and the generalized Pólya–Aeppli process. The state probabilities of these compound fractional Poisson processes solve a system of fractional differential equations that involves the Caputo fractional derivative of order 0<β<1. It is shown that these processes are limiting cases of a recently introduced process, namely, the generalized counting process. We obtain the mean, variance, covariance, long-range dependence property, etc., for these processes. Further, we obtain several equivalent forms of the one-dimensional distribution of fractional versions of these processes.

Fractional Criticality Theory and Its Application in Seismology

“…We consider the compound Poisson process [1,2] in its fractional generalization [3][4][5] with a power-law distribution of events' recurrence frequencies [6,7]. Due to scaling and hereditarity, this process has all the necessary properties with which to describe critical phenomena, including seismic ones.…”

“…In the next section, we will define the compound fractional Poisson process [3][4][5] and provide it with a power-law of the distribution of the amplitudes of changes, i.e., scaling [6,7], and thus obtain the power-law compound fractional Poisson process. In Section 3, we will define the critical indices and regimes of this process, which will allow us to consider it as a fairly universal model of the hereditarian theory of criticality.…”