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: $$\phi(\theta)=\lambda\left[\lambda+\theta+k\int_0^\infty\left(1-e^{-\theta z}\right)\,dG(z)\right]^{-1},$$ for some positive real numbers $\lambda, k$, and some distribution function $G$ with $G(\infty)=1$. The intensity process $\Lambda(t)$ of the corresponding DSPP jumps between $\lambda$ and $0$, with the time spent at $\lambda$ 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 $G$.