Quasi-stationary Distributions: A Bibliography

Quasi-stationary distributions have been used to model the long-term behaviour of stochastic systems which in some sense terminate, but appear to be stationary over any reasonable time scale. Imagine population is observed to be extant at some time t. What is the chance of there being precisely i individuals present? If we were equipped with the full set of state probabilities, we would evaluate the probability ui(t) of the being i individuals present conditional on their being at least 1. It would then be natural for us to seek a distribution (ui, i ∈ S) over the set of extant states S such that if ui(s) = ui for a particular s > 0, then ui(t) = ui for all t > s. Such a distribution is called a stationary conditional distribution or quasi-stationary distribution. This distribution might then also be a limiting conditional distribution in that ui(t)→ ui as t→∞, and thus be of use in modelling the long-term behaviour of the process. Yaglom [453] was the first to identify explicitly a limiting conditional distribution, establishing the existence of such for the subcritical Bienaymé-Galton-Watson branching process. However, the idea of a limiting conditional distribution goes back much further, at least to Wright [449, Page 111] in his discussion of gene frequencies in finite populations. The idea of “quasi stationarity” was crystalized by Bartlett [32, Page 38] and he later coined the term “quasi-stationary distribution” [33, Page 24]. But, it was not until the early sixties, and largely stimulated by the remarkable work of Vere-Jones [440], and later Kingman [221], Darroch and Seneta [99], Seneta and Vere-Jones [385], and Darroch and Seneta [100], that a general theory was annunciated. Since then, quasi-stationary distributions have appeared in a variety of diverse contexts, including chemical reaction kinetics, reliability theory, genetics, epidemics, ecology, finance, and telecommunications, and this work has stimulated further developments in the theory. Modern key papers in the area are Ferrari, Kesten, Mart́ınez and Picco [124] and Kesten [201]. I present here a bibliography of work on quasi-stationary distributions. This includes work on quasi-stationary distributions per se (stationary conditional distributions), limiting conditional distributions (often called quasi-stationary distributions, and also called Yaglom limits and quasi-limiting distributions), the companion topics of geometric and exponential ergodicity, R-classification of states and R-invariant measures (et cetera), ratio limit theorems, analysis of processes conditioned to stay within a given region (particularly weak convengence of those processes), and papers dealing with diffusion approximations which specifically describe quasi stationarity of evanescent processes. Published work is cited under various headings. Several works appear under more than one heading. The final section lists the same works in chronological order.