Minimal quasi-stationary distribution approximation for a birth and death process

TL;DR

This paper develops a criterion for positive recurrence in birth-death processes and demonstrates that a Fleming-Viot particle system's empirical distribution converges to the minimal quasi-stationary distribution.

Contribution

It introduces a Lyapunov-type criterion for recurrence and proves convergence of a particle system to the minimal quasi-stationary distribution.

Findings

Lyapunov criterion for positive recurrence established

Convergence of empirical distributions to the minimal quasi-stationary distribution proven

New insights into the domain of attraction for quasi-stationary distributions

Abstract

In a first part, we prove a Lyapunov-type criterion for the $\xi\_1$-positive recurrence of absorbed birth and death processes and provide new results on the domain of attraction of the minimal quasi-stationary distribution. In a second part, we study the ergodicity and the convergence of a Fleming-Viot type particle system whose particles evolve independently as a birth and death process and jump on each others when they hit $0$. Our main result is that the sequence of empirical stationary distributions of the particle system converges to the minimal quasi-stationary distribution of the birth and death process.