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

Abstract

For downward skip-free continuous-time Markov chains on non-negative integers killed at zero, the existence of the quasi-stationary distribution is studied. The scale function for the process 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.