Quasi-Stationary Distributions for Reducible Absorbing Markov Chains in Discrete Time

We consider discrete-time Markov chains with one coffin state and a finite set S of transient states, and are interested in the limiting behaviour of such a chain as time n tends to infinity, conditional on survival up to n. It is known that, when S is irreducible, the limiting conditional distribution of the chain equals the (unique) quasi-stationary distribution of the chain, while the latter is the (unique) ρ-invariant distribution for the one-step transition probability matrix of the (sub)Markov chain on S, ρ being the Perron–Frobenius eigenvalue of this matrix. Addressing similar issues in a setting in which S may be reducible, we identify all quasi-stationary distributions and obtain a necessary and sufficient condition for one of them to be the unique ρ-invariant distribution. We also reveal conditions under which the limiting conditional distribution equals the ρ-invariant distribution if it is unique. We conclude with some examples.