Renewal Processes Represented as Doubly Stochastic Poisson Processes

2024/09/27 by Du, Xinlong, Honnappa, Harsha
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2409.18362

Abstract

This paper gives an elementary proof for the following theorem: a renewal process can be represented by a doubly-stochastic Poisson process (DSPP) if and only if the Laplace-Stieltjes transform of the inter-arrival times is of the following form: ϕ(θ)=λ[λ+θ+k∫₀^∞(1-e^-θz) dG(z)]^-1, for some positive real numbers λ, k, and some distribution function G with G(∞)=1. The intensity process Λ(t) of the corresponding DSPP jumps between λ and 0, with the time spent at λ being independent random variables that are exponentially distributed with mean 1/k, and the time spent at 0 being independent random variables with distribution function G.

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