Quasistationary Distributions for Continuous Timemarkov Chains

Recently, Elmes, Pollett and Walker 2] proposed a deenition of a quasistationary distribution to accommodate absorbing Markov chains for which absorption occurs with probability less than 1. We will show that the probabilistic interpretation pertaining to cases where absorption is certain (van Doorn 13]) does not hold in the present context. We prove that the state probabilities at time t conditional on absorption taking place after t generally depend on t. Conditions are derived under which there is no initial distribution such that the conditional state probabilities are stationary. 1. Quasistationary distributions Let (X(t); t 0) be a continuous-time Markov chain with state space S = f0gC, where C = f1; 2; : : : g is an irreducible class and 0 is an absorbing state. Let P() = (p ij (); i; j 2 S), where p ij (t) = Pr(X(t) = jjX(0) = i), t > 0, be the transition function of the chain, standard in the sense that p ij (0+) = ij , and suppose that for some i 2 C, p i0 (t) > 0 for some (and then all) t > 0. For simplicity we shall suppose that P is honest. The deenition of a quasistationary distribution which was introduced by van Doorn 13] is as follows: Deenition 1 Let m = (m j ; j 2 C) be a probability distribution over C and let p j (t) = X i2C m i p ij (t); j 2 S; t > 0: Then, m is a quasistationary distribution if, for all t > 0 and j 2 C, p j (t) P i2C p i (t) = m j : The probabilistic interpretation is obvious: m is a quasistationary distribution if the conditional state probabilities, Pr(X(t) = jjX(t) 2 C), j 2 C, are the same for all t.