The criterion for uniqueness of quasi-stationary distributions of Markov processes and their domain of attraction problem

TL;DR

This paper establishes conditions for the existence and uniqueness of quasi-stationary distributions in Markov processes with absorbing states, proving a conjecture and improving existing theorems related to extinction times.

Contribution

It provides new criteria for the uniqueness of quasi-stationary distributions and confirms van Doorn's conjecture, advancing understanding of the domain of attraction for such processes.

Findings

Identified five equivalent conditions for uniformly bounded extinction times

Proved the existence and uniqueness of quasi-stationary distributions under these conditions

Improved previous theorems related to extinction time behavior

Abstract

We consider a Markov process $ X(t) $ on the nonnegative integers $E= S \cup \{0\}$, where $S=\{1,2,...\}$ is an irreducible class and 0 is an absorbing state. In this paper, we investigate conditions under which the quasi-stationary distribution for $X(t)$ exists and is unique, and any initial distribution supported in $S$ is in the domain of attraction of this quasi-stationary distribution. We further find five conditions which are equivalent to that the extinction time is uniformly bounded. As a consequence, we prove the van Doorn's conjecture in \cite{VD2012}. And we can greatly improve theorem 1 in \cite{VD2012}.