On the moment distance of Poisson processes

TL;DR

This paper derives a closed-form analytical formula for the expected power of the distance between two independent Poisson processes' arrival times, revealing explicit identities involving gamma functions and Pochhammer polynomials.

Contribution

It provides the first explicit formulas for the expected moments of the distance between two independent Poisson processes' arrival times.

Findings

Derived a closed-form formula for expected distance moments.

Expressed the expectation using Pochhammer polynomials and gamma functions.

Special case for r=0 yields a simple gamma function identity.

Abstract

Consider the distance between two i.i.d. and independent Poisson processes with arrival rate $\lambda>0$ and respective arrival times $X_1,X_2,\dots$ and $Y_1,Y_2,\dots$ on a line. We give a closed analytical formula for the %expected distance to the power $a$ $\E{|X_{k+r}-Y_k|^a}, $ for any integer $k\ge 1, r\ge 0$ and $a\ge 1.$ The expected difference of the arrival times to the power $a$ between two i.i.d. and independent Poisson processes we represent as the combination of the Pochhammer polynomials. Especially, for $r=0$ and any positive integer $a,$ the following identity is valid $$ \E{|X_k-Y_k|^a}=\frac{a!}{\lambda^a}\frac{\Gamma\left(\frac{a}{2}+k\right)}{\Gamma(k)\Gamma\left(\frac{a}{2}+1\right)}, $$ where $\Gamma(z)$ is Gamma function.