Representation and Characterization of Quasistationary Distributions for Markov Chains

Abstract: This work provides 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 under the law of the process with the QSD as the initial distribution. The analysis associated with the 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, the finite or infinite moment generating function regimes, respectively. For parameters in the finite regime, it is shown that when exist, all QSDs are in the convex cone of a Martin entry boundary associated with the parameter. The infinite regime corresponds to at most one parameter value and at most one QSD. In this regime, when a QSD exists, it is unique and can be represented by a renewal-type formula. Multiple applications to the findings are presented, including revisiting some of the main classical results in the area.