On the moment distance of Poisson processes

2015/07/03 by Rafał Kapelko, Kapelko, Rafał
Mathematics · Business, Management and Accounting · Computer Science · #Advanced Combinatorial Mathematics #Advanced Queuing Theory Analysis #Optimization and Search Problems

paper · pdf · doi:10.48550/arxiv.1507.01048

Abstract

Consider the distance between two i.i.d. and independent Poisson processes with arrival rate λ>0 and respective arrival times X₁,X₂,… and Y₁,Y₂,… 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≥ 1, r≥ 0 and a≥ 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=(a!)/(λ^a)(Γ((a)/(2)+k))/(Γ(k)Γ((a)/(2)+1)), where Γ(z) is Gamma function.

Related

• Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and U-statistics
• Inter-arrival time distribution for the non-homogeneous Poisson process
• Tight Lower Bounds on the Contact Distance Distribution in Poisson Hole Process
• On probabilities associated with the minimum distance between events of a Poisson process in a finite interval, and Erratum/Addendum to it
• Stein's method and Papangelou intensity for Poisson or Cox process approximation