Representation and characterization of quasistationary distributions for Markov chains

This work provides a complete description of Quasistationary Distributions (QSDs) for Markov chains with a unique absorbing state and an irreducible set of non-absorbing states. As is well-known, every QSD has an associated absorption parameter describing the exponential tail of the absorption time with the QSD as the initial distribution. Our analysis of existence and representation of QSDs corresponding to a given parameter is according to whether the moment generating function of the absorption time starting from any non-absorbing state evaluated at the parameter is finite or infinite, representing the finite or infinite moment generating function regimes, respectively. For absorption parameters in the finite regime, all QSDs are in the convex cone of a Martin entrance boundary associated with the parameter. The infinite regime corresponds to at most one absorption parameter value. In this regime, when a QSD exists, it is unique and can be represented by a renewal-type formula. Multiple applications are presented, including revisiting some of the main classical results in the area.