On the Integral of Fractional Poisson Processes

Abstract

In this paper we consider the Riemann--Liouville fractional integral ${\mathcal{N}}^{\alpha ,\nu }(t)=\frac{1}{\Gamma (\alpha )}{\int }_0^t(t-s{)}^{\alpha -1}{N}^{\nu }(s)\mathrm{d}s$, where $N^{\nu }(t)$, $t\ge 0$, is a fractional Poisson process of order $\nu \in (0,1]$, and $\alpha >0$. We give the explicit bivariate distribution $\mathrm{Pr}\{N^{\nu }(s)=k,N^{\nu }(t)=r\}$, for $t\ge s$, $r\ge k$, the mean $\mathbb{E}{\mathcal{N}}^{\alpha ,\nu }(t)$ and the variance $\mathbb{V}\text{ar}{\mathcal{N}}^{\alpha ,\nu }(t)$. We study the process ${\mathcal{N}}^{\alpha ,1}(t)$ for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on ${\mathcal{N}}^{1,1}(t)$ are presented in the last section where also distributional properties of the integrated Poisson process (including the representation as random sums) is derived. The integral of powers of the Poisson process is examined and its connections with generalised harmonic numbers is discussed.

References in corpus (3)

• The Space-Fractional Poisson Process
• Full characterization of the fractional Poisson process
• Renewal processes based on generalized Mittag--Leffler waiting times

Cited by in corpus (5)

• Hilfer-Prabhakar Derivatives and Some Applications
• Fractional Poisson processes and their representation by infinite systems of ordinary differential equations
• Properties of Poisson processes directed by compound Poisson-Gamma subordinators
• Regulating stochastic clocks
• Some Time-changed fractional Poisson processes