On the Integral of Fractional Poisson Processes

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 ds $, 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 $\Pr \{ N^\nu(s)=k, N^\nu(t)=r \}$, 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.