Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically

TL;DR

This paper investigates the properties of quasi-stationarity and quasi-ergodicity in discrete-time Markov chains with moving absorbing boundaries, revealing limitations with oscillating boundaries and establishing conditions for quasi-ergodicity.

Contribution

It demonstrates the non-existence of quasi-stationary distributions with oscillating boundaries and proves the existence of quasi-ergodic distributions for fixed and moving boundaries.

Findings

Quasi-stationary distributions are not well-defined with oscillating boundaries.

Quasi-ergodic distributions exist for fixed boundary Markov chains.

Quasi-ergodicity can be extended to moving boundary cases using fixed boundary results.

Abstract

We are interested in quasi-stationarity and quasi-ergodicity when the absorbing boundary is moving. First we show that, in the moving boundary case, the quasi-stationary distribution and the quasi-limiting distribution are not well-defined when the boundary is oscillating periodically. Then we show the existence of a quasi-ergodic distribution for any discrete-time irreducible Markov chain defined on a finite space state in the fixed boundary case. Finally we use this last result to show the quasi-ergodicity in the moving boundary case.