Renewal Processes Represented as Doubly Stochastic Poisson Processes

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: $$\ddot{I}•(\hat{I}¸)=\hat{I}»{\left[\hat{I}»+\hat{I}¸+k{\int }_0^{\infty }\left(1-e^{-\hat{I}¸z}\right)dG(z)\right]}^{-1},$$ for some positive real numbers $\hat{I}»,k$, and some distribution function $G$ with $G(\infty )=1$. The intensity process $\hat{I}›(t)$ of the corresponding DSPP jumps between $\hat{I}»$ and $0$, with the time spent at $\hat{I}»$ 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 $G$.