Asymptotic results for a multivariate version of the alternative fractional Poisson process

TL;DR

This paper investigates the asymptotic behavior of a multivariate fractional Poisson process with specific marginal distributions, extending previous univariate results and exploring statistical estimation of the fractional parameter.

Contribution

It introduces a new multivariate fractional Poisson process with specified marginal distributions and derives large and moderate deviation results, extending prior univariate findings.

Findings

Established large deviation principles for the multivariate process.

Derived moderate deviation results for the process.

Provided statistical methods for estimating the fractional parameter.

Abstract

A multivariate fractional Poisson process was recently defined in Beghin and Macci (2016) by considering a common independent random time change for a finite dimensional vector of independent (non-fractional) Poisson processes; moreover it was proved that, for each fixed $t\geq 0$, it has a suitable multinomial conditional distribution of the components given their sum. In this paper we consider another multivariate process $\{\underline{M}^\nu(t)=(M_1^\nu(t),\ldots,M_m^\nu(t)):t\geq 0\}$ with the same conditional distributions of the components given their sums, and different marginal distributions of the sums; more precisely we assume that the one-dimensional marginal distributions of the process $\left\{\sum_{i=1}^mM_i^\nu(t):t\geq0\right\}$ coincide with the ones of the alternative fractional (univariate) Poisson process in Beghin and Macci (2013). We present large deviation results for $\{\underline{M}^\nu(t)=(M_1^\nu(t),\ldots,M_m^\nu(t)):t\geq 0\}$, and this generalizes the result in Beghin and Macci (2013) concerning the univariate case. We also study moderate deviations and we present some statistical applications concerning the estimation of the fractional parameter $\nu$.