Quasi-stationary distributions for single death processes with killing

TL;DR

This paper investigates the existence, uniqueness, and convergence properties of quasi-stationary distributions for single death processes with killing, using probabilistic and potential theoretic methods.

Contribution

It establishes conditions for the existence and uniqueness of quasi-stationary distributions in death processes with killing, and characterizes their exponential convergence.

Findings

Existence and uniqueness of quasi-stationary distributions under certain conditions

Conditions for exponential convergence in total variation norm

Application of Doob's $h$-transform and potential theory

Abstract

This paper studies the quasi-stationary distributions for a single death process (or downwardly skip-free process) with killing defined on the non-negative integers, corresponding to a non-conservative transition rate matrix. The set $\{1,2,3,\cdots\}$ constitutes an irreducible class and $0$ is an absorbing state. For the single death process with three kinds of killing term, we obtain the existence and uniqueness of the quasi-stationary distribution. Moreover, we derive the conditions for exponential convergence to the quasi-stationary distribution in the total variation norm. Our main approach is based on the Doob's $h$-transform, potential theory and probabilistic methods.