The Annals of Probability 2001, Vol. 29, No. 1, 437–446 ON THE EXISTENCE OF A QUASISTATIONARY MEASURE FOR A MARKOV CHAIN By Jean B. Lasserre and Charles E. M. Pearce LAAS-CNRS and University of Adelaide We consider a Markov chain on a locally compact metric space with an absorbing set. Necessary and sufficient conditions are provided for the existence of a quasistationary probability distribution. 1. Introduction. Consider a Markov chain  on a locally compact sep- arable metric space X with an absorbing set S ⊂ X; that is, once in S the chain  remains in S with probability 1. As absorption can take a long time, one is often interested in the evolution of the distribution of  conditional on absorption not yet having taken place. This issue has been investigated in the pioneering papers of Seneta and Vere-Jones [13] and others for countable state spaces. For a review, see [11]. It has been shown that under various conditions this conditional probability has a limit distribution, which is called a quasistationary distribution or QSD for short. For instance, it was shown that the existence of the Yaglom limit for some initial state x implies the existence of a QSD. For accounts of limiting conditional distributions, the reader is referred to [8] and [14]–[16]. More recently, still in the countable case and in continuous time, Ferrari, Kesten, Martinez and Picco [4] have also proved the existence of a QSD using renewal arguments and under assumptions on the distribution of the absorp- tion time. Finally, in a recent paper, Hognas [7] considered a parametrized single-species population model of the Ricker type and proved the existence of a QSD under easily checked assumptions on the model. He then analyzed the asymptotic behavior of the QSD as the parameter γ vanishes. Interestingly enough, two assumptions in [7] and [4] are quite opposite. Hognas [7] assumes that the one-step probability of absorption goes to unity as the distance to the absorbing set becomes large, whereas the discrete-time version of one condition in Ferrari, Kesten, Martinez and Picco [4] implies that this one-step probability vanishes as the state becomes large! The former hypothesis is particular to population growth models. In all of the previously cited works, the state space was countable and the arguments for proving the existence of a QSD dependent on the discrete nature of X. Seneta and Vere-Jones [13] and Hognas [7] used the Perron–Frobenius theory of nonnegative matrices. Ferrari, Kesten, Martinez and Picco [4] made elegant use of renewal arguments and fixed-point techniques and it is possible that their approach can be extended to the present context. However, as acknowledged by the authors, the conditions are annoying for they restrict Received May 1998; revised May 2000. AMS 2000 subject classifications. Primary 60J05; secondary 28A33. Key words and phrases. Markov chains, Borel spaces, quasistationarity. 437