Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

Abstract

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

Geomatric ergodicity and quasi-stationarity in discrete-time birth-death processes

“…The following sufficient condition for strict inequality to hold in (2.16), partly noted earlier by Karlin and McGregor [15], can also be obtained from the Lemmas 2.3 and 2.4. [7] Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes 127…”

“…Further information on the random walk measure x\r can be obtained by transforming the sequence of random walk polynomials into a sequence of birth-death polynomials, as suggested by Karlin and McGregor [15]. In this way many results from the theory of birth-death processes in continuous time (see Karlin and McGregor [13], [14] and van Doom [5], [7]) can be translated in terms of random walks. Pursuing this approach we define 20) and…”

“…https://doi.org/10.1017/S0334270000007621 those for rj by a simple transformation of the results in Lemma 3.1 and its corollary. Indeed, it is readily seen that the polynomials {(-1)-' Qj(-x)}j satisfy the recurrence (2.5) with rj replaced by -r 7 . Hence, replacing rj by -£ and r, by -r, in Lemma 3.1 and its corollary gives us representations and bounds for £, see van Doom [6].…”

“…The present paper constitutes to some extent the discrete-time counterpart of van Doom [5] and van Doom [7], which discuss exponential ergodicity and quasistationarity, respectively, in continuous-time birth-death processes. However, a study of limiting conditional (or quasi-limiting) distributions as in [7], which turned out to be considerably more complicated in discrete time, has been relegated to a separate paper, see van Doom and Schrijner [9].…”

“…However, a study of limiting conditional (or quasi-limiting) distributions as in [7], which turned out to be considerably more complicated in discrete time, has been relegated to a separate paper, see van Doom and Schrijner [9].…”