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.
taking values in
There is a measurable set
the hitting time of
, also called killing time.
The general definition[1] is: a probability measure
is said to be a quasi-stationary distribution (QSD) if for every measurable set
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:[1][2] if
is a QSD then there exists
θ ( ν ) > 0
> t ) = exp ( − θ ( ν ) × t )
ϑ < θ ( ν )
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.
A necessary condition for the existence of a QSD is
θ ( ν )
ϑ = θ ( ν )
because other wise this would lead to the contradiction
θ ( ν )
θ ( ν )
A sufficient condition for a QSD to exist is given considering the transition semigroup
is a compact Hausdorff space and that
preserves the set of continuous functions, i.e.
, there exists a QSD.
The works of Wright on gene frequency in 1931[3] and of Yaglom on branching processes in 1947[4] already included the idea of such distributions.
The term quasi-stationarity applied to biological systems was then used by Bartlett in 1957,[5] who later coined "quasi-stationary distribution".
[6] Quasi-stationary distributions were also part of the classification of killed processes given by Vere-Jones in 1962[7] and their definition for finite state Markov chains was done in 1965 by Darroch and Seneta.
[8] Quasi-stationary distributions can be used to model the following processes: