On the quasi-ergodic distribution of absorbing Markov processes

TL;DR

This paper establishes conditions for the existence of quasi-ergodic distributions in absorbing Markov processes, specifically applying an orthogonal-polynomial approach to birth-death processes and comparing quasi-ergodic and quasi-stationary distributions.

Contribution

It provides a sufficient condition for quasi-ergodic distribution existence and demonstrates the relationship between quasi-ergodic and quasi-stationary distributions in birth-death processes.

Findings

Quasi-ergodic distribution exists under specified conditions.

Orthogonal-polynomial approach is effective for birth-death processes.

Quasi-ergodic distribution is stochastically larger than the quasi-stationary distribution.

Abstract

In this paper, we give a sufficient condition for the existence of a quasi-ergodic distribution for absorbing Markov processes. Using an orthogonal-polynomial approach, we prove that the previous main result is valid for the birth-death process on the nonnegative integers with 0 an absorbing boundary and $\infty$ an entrance boundary. We also show that the quasi-ergodic distribution is stochastically larger than the unique quasi-stationary distribution in the sense of monotone likelihood-ratio ordering for the birth-death process.