On the integral of fractional Poisson processes

In this paper, we consider the Riemann–Liouville fractional integral Nα,ν(t)=1Γ(α)∫0t(t−s)α−1Nν(s)ds, where Nν(t)Nν(t), t≥0t≥0, is a fractional Poisson process of order ν∈(0,1]ν∈(0,1], and α>0α>0. We give the explicit bivariate distribution Pr{Nν(s)=k,Nν(t)=r}Pr{Nν(s)=k,Nν(t)=r}, for t≥st≥s, r≥kr≥k, the mean ENα,ν(t) and the variance VarNα,ν(t). We study the process Nα,1(t)Nα,1(t) for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on N1,1(t)N1,1(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 generalized harmonic numbers are discussed.