Existence of quasi-stationary distributions for downward skip-free Markov chains

TL;DR

This paper investigates the conditions under which downward skip-free continuous-time Markov chains on non-negative integers have quasi-stationary distributions, extending Feller's boundary classification using a new scale function approach.

Contribution

It introduces a scale function to classify boundaries and characterizes the existence of quasi-stationary distributions for these Markov chains.

Findings

Boundary classification depends on a new integrability condition on the scale function.

Existence of quasi-stationary distributions is fully characterized by the boundary classification.

Provides an extension of Feller's boundary classification for birth-and-death processes.

Abstract

For downward skip-free continuous-time Markov chains on non-negative integers stopped at zero, existence of a quasi-stationary distribution is studied. The scale function for these processes is introduced and the boundary is classified by a certain integrability condition on the scale function, which gives an extension of Feller's classification of the boundary for birth-and-death processes. The existence and the set of quasi-stationary distributions are characterized by the scale function and the new classification of the boundary.