Generalized Counting Processes in a Stochastic Environment

Davide Cocco,Massimiliano Giona

doi:10.3390/math9202573

Abstract

This paper addresses the generalization of counting processes through the age formalism of Lévy Walks. Simple counting processes are introduced and their properties are analyzed: Poisson processes or fractional Poisson processes can be recovered as particular cases. The stationarity assumption in the renewal mechanism characterizing simple counting processes can be modified in different ways, leading to the definition of generalized counting processes. In the case that the transition mechanism of a counting process depends on the environmental conditions—i.e., the parameters describing the occurrence of new events are themselves stochastic processes—the counting processes is said to be influenced by environmental stochasticity. The properties of this class of processes are analyzed, providing several examples and applications and showing the occurrence of new phenomena related to the modulation of the long-term scaling exponent by environmental noise.

Highlights

A counting process is nothing but a stochastic process N (t), t ≥ 0 that counts the number of events that have occurred up to the current time t, equipped with the following assumptions [1,2]: N (0) = 0;N (t) ∈ 0, 1, 2, .., ∀t ∈ R+ ; for 0 < t < t0, N (t0 ) − N (t) is the number of events occurring in the interval
This paper addresses the generalization of counting processes through the age formalism of Lévy Walks
In the case that the transition mechanism of a counting process depends on the environmental conditions—i.e., the parameters describing the occurrence of new events are themselves stochastic processes—the counting processes is said to be influenced by environmental stochasticity

Summary

Introduction

A counting process is nothing but a stochastic process N (t), t ≥ 0 that counts the number of events that have occurred up to the current time t, equipped with the following assumptions [1,2]:. Many contributions have been focused on the generalization of the standard Poisson process using fractional calculus and fractional operators providing a fractional version of the Poisson process that allows a power law decay of the counting probabilities to be predicted [10,11,12,13] A representative example of application of this class of processes is the power-law decay of the duration of network sessions at large session-times. This leads to the presence of two different transition ages related to the occurrence of events and to transitions in the environmental fluctuations

Simple Counting Processes

Generalized Counting Processes

Counting Processes in a Stochastic Environment

Scaling Analysis

Extensions

Concluding Remarks