In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for a long time without reaching it. The most common example is the evolution of a population: the only equilibrium is when there is no one left, but if we model the number of people it is likely to remain stable for a long period of time before it eventually collapses.
Formal definition
We consider a Markov process taking values in. There is a measurable set of absorbing states and. We denote by the hitting time of, also called killing time. We denote by the family of distributions where has original condition. We assume that is almost surely reached, i.e.. The general definition is: a probability measure on is said to be a quasi-stationary distribution if for every measurable set contained in, where. In particular
General results
Killing time
From the assumptions above we know that the killing time is finite with probability 1. A stronger result than we can derive is that the killing time is exponentially distributed: if is a QSD then there exists such that. Moreover, for any we get.
Existence of a quasi-stationary distribution
Most of the time the question asked is whether a QSD exists or not in a given framework. From the previous results we can derive a condition necessary to this existence. Let. A necessary condition for the existence of a QSD is and we have the equality Moreover, from the previous paragraph, if is a QSD then. As a consequence, if satisfies then there can be no QSD such that because other wise this would lead to the contradiction. A sufficient condition for a QSD to exist is given considering the transition semigroup of the process before killing. Then, under the conditions that is a compact Hausdorff space and that preserves the set of continuous functions, i.e., there exists a QSD.
History
The works of Wright. on gene frequency in 1931 and Yaglom on branching processes in 1947 included already the idea of such distributions. The term quasi-stationarity applied to biological systems was then used by Barlett in 1957, who later coined "quasi-stationary distribution" in. Quasi-stationary distributions were also part of the classification of killed processes given by Vere-Jones in 1962 and their definition for finite stateMarkov chains was done in 1965 by Darroch and Seneta
Examples
Quasi-stationary distributions can be used to model the following processes:
Evolution of a population by the number of people: the only equilibrium is when there is no one left.
Evolution of a contagious disease in a population by the number of people ill: the only equilibrium is when the disease disappears.
Transmission of a gene: in case of several competing alleles we measure the number of people who have one and the absorbing state is when everybody has the same.
Voter model: where everyone influences a small set of neighbors and opinions propagate, we study how many people vote for a particular party and an equilibrium is reached only when the party has no voter, or the whole population voting for it.
