On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains

Abstract

The time to absorption from the set T of transient states of a Markov chain may be sufficiently long for the probability distribution over T to settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

Quasi-stationarity and quasi-ergodicity of general Markov processes

“…There are many results on the existence and uniqueness of quasi-stationary and quasi-egodic distributions. When the state space E of the Markov process is countable, quasi-stationarity and quasi-ergodicity have been thoroughly studied, see, for instance, [7,8,10,23]. For Markov processes on general state spaces, Breyer and Roberts [2] established the existence and uniqueness of quasi-ergodic distributions under the assumption that the Markov process is positive λ-recurrent for some constant λ ≤ 0 (see also the recent paper [3,18]).…”

On the quasi-stationary distribution for queueing networks with defective routing

“…is primitive, Darroch and Seneta [2] show that the quasi-stationary distribution can be obtained as the unique Perron-Frobenius left eigenvector of Q, that is, with a < 1 the maximal real eigenvalue of Q (a < 1 because Q is substochastic), it must be that jr is the unique solution of…”

“…When the time to absorption is sufficiently long, under some conditions, this distribution is known as the quasi-stationary distribution. The theory for quasi-stationary distributions of Markov chains with finite [2] On the quasi-stationary distribution for queueing networks with defective routing 455 state space is well-developed. The product-form results of this note can be seen as examples of the results in the papers of Darroch and Seneta [2], [3].…”

“…The theory for quasi-stationary distributions of Markov chains with finite [2] On the quasi-stationary distribution for queueing networks with defective routing 455 state space is well-developed. The product-form results of this note can be seen as examples of the results in the papers of Darroch and Seneta [2], [3]. Further results on quasi-stationary distributions and related notions such as R-recurrence can, for example, be found in Vere-Jones [8], Seneta and Vere-Jones [7], Kesten [5].…”

Semi-Markov processes on a general state space: α-theory and quasi-stationarity

“…For Markov chains in discrete time, and with a countable state space, the analytic behaviour ('R-theory') of the transition probabilities is investigated by Vere-Jones (1962) and(1967), and the application to quasi-stationary problems is given by Seneta and Vere-Jones (1966), following the initial work by Darroch and Seneta (1965) on quasi-stationarity for finite state space chains. The R-theory and quasi-stationarity for general state space is developed by Tweedie (1974a) and.…”