Approximation of birth–death processes

A birth–death process is a special type of continuous-time Markov chains with minimal state space $N$. Its resolvent matrix can be fully characterized by a set of parameters $(\gamma ,\beta ,\nu )$, where $\gamma$ and $\beta$ are non-negative constants, and $\nu$ is a positive measure on $N$. By employing the Ray-Knight compactification, the birth–death process can be realized as a càdlàg process with strong Markov property on the one-point compactification space ${\overline{N}}_{\partial }$, which includes an additional cemetery point $\partial$. In a certain sense, the three parameters that determine the birth–death process correspond to its killing, reflecting, and jumping behaviors at $\infty$ used for the one-point compactification, respectively.

In general, providing a clear description of the trajectories of a birth–death process, especially in the pathological case where $\left|\nu \right|=\infty$, is challenging. This paper aims to address this issue by studying the birth–death process using approximation methods. Specifically, we will approximate the birth–death process with simpler birth–death processes that are easier to comprehend. For two typical approximation methods, our main results establish the weak convergence of a sequence of probability measures, which are induced by the approximating processes, on the space of all càdlàg functions. This type of convergence is significantly stronger than the convergence of transition matrices typically considered in the theory of continuous-time Markov chains.