Correlated fractional counting processes on a finite time interval

TL;DR

This paper introduces correlated fractional counting processes on finite intervals, generalizing existing models, and explores their properties, including space-time fractional Poisson and negative binomial processes, highlighting differences from prior models.

Contribution

It presents a generalized framework for correlated fractional counting processes, extending previous models and analyzing their multivariate distributions and correlations.

Findings

Multivariate distributions differ from classical models when correlation is introduced.

The processes include space-time fractional Poisson and fractional negative binomial processes.

The models reduce to known processes when the correlation parameter is zero.

Abstract

We present some correlated fractional counting processes on a finite time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to zero, the univariate distributions coincide with the ones of the space-time fractional Poisson process in Orsingher and Polito (2012). On the other hand, when we consider the time fractional Poisson process, the multivariate finite dimensional distributions are different from the ones presented for the renewal process in Politi et al. (2011). Another case concerns a class of fractional negative binomial processes.