ON THE EXISTENCE OF QUASI-STATIONARY DISTRIBUTIONS IN DENUMERABLE R-TRANSIENT MARKOV-CHAINS

Let {X(n), n = 0, 1, 2,...} be a transient Markov chain which, when restricted to the state space N + = {1, 2,...}, is governed by an irreducible, aperiodic and strictly substochastic matrix P = (p(ij)), and let p(ij)(n) = Pr[X(n) = j, X(k) is-an-element-of N + for k = 0, 1,..., n\X0 = i], i, j is-an-element-of N +. The prime concern of this paper is conditions for the existence of the limits, q(ij) say, of q(ij)(n) = P(ij)(n)/SIGMA(j = 1) infinity p(ij)(n) as n --> infinity. If SIGMA(j = 1) infinity q(ij) = 1, the distribution (q(ij)) is called the quasi-stationary distribution of {X(n)} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vector x = (x(i)) satisfying rx(T) = x(T)P and SIGMA(i = 1) infinity X(i) = 1 exists, where r is the convergence norm of P, i.e. r = R-1 and R = sup{z: SIGMA(n = 1) infinity P(n)z(n) < infinity}, and T denotes transpose, then it is unique, positive elementwise, and q(ij)(n) necessarily converge to x(j) as n --> infinity. Unlike existing results in the literature, our results can be applied even to the R-null and R-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix is R-transient is discussed to demonstrate the usefulness of our results.