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 $G$.

翻译：本文给出了以下定理的一个基本证明：当且仅当到达间隔时间的拉普拉斯-斯蒂尔切斯变换具有以下形式时，一个更新过程可以表示为一个双重随机泊松过程（DSPP）： $$\phi(\theta)=\lambda\left[\lambda+\theta+k\int_0^\infty\left(1-e^{-\theta z}\right)\,dG(z)\right]^{-1},$$ 其中 $\lambda, k$ 为正实数，$G$ 为满足 $G(\infty)=1$ 的某个分布函数。对应 DSPP 的强度过程 $\Lambda(t)$ 在 $\lambda$ 和 $0$ 之间跳跃，停留在 $\lambda$ 的时间是独立的随机变量，服从均值为 $1/k$ 的指数分布，而停留在 $0$ 的时间是独立的随机变量，其分布函数为 $G$。