On the integral of fractional Poisson processes

2012/12/26 by Enzo Orsingher, Federico Polito · 19 citations
Mathematics · #Combinatorics #Fractional Differential Equations Solutions #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Order (exchange) #Poisson distribution #Poisson kernel #Poisson process #Statistical Distribution Estimation and Applications #Statistics #math.PR

paper · pdf · doi:10.1016/j.spl.2012.12.016

published in Statistics & Probability Letters 83(4), 1006-1017 (Elsevier BV)

openalex publication_date 2012/12/26 · arxiv created 2013/03/26 · arxiv updated 2014/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper we consider the Riemann--Liouville fractional integral N^α,ν(t)= (1)/(Γ(α)) ∫₀^t (t-s)^α-1N^ν(s) \mathrm ds , where N^ν(t), t ≥ 0, is a fractional Poisson process of order ν∈ (0,1], and α> 0. We give the explicit bivariate distribution Pr \N^ν(s)=k, N^ν(t)=r \, for t ≥ s, r ≥ k, the mean 𝔼 N^α,ν(t) and the variance \mathbbVar N^α,ν(t). We study the process N^α,1(t) for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on 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.

Citations

• Renewal processes based on generalized Mittag--Leffler waiting times
• Fractional Poisson Process
• The Space-Fractional Poisson Process

Cited by

• Point processes of the Poisson-Skellam family