Quasi-stationary distribution explained

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

(Y _{t)} _{t} taking values in l{X} . There is a measurable set l{X} ^{tr} of absorbing states and l{X} ^{a} = l{X} \setminus l{X} ^{\operatorname{tr} }. We denote by T the hitting time of l{X} ^{\operatorname{tr} }, also called killing time. We denote by \{ \operatorname{P} _{x} \mid x \in l{X} \} the family of distributions where \operatorname{P} _{x} has original condition Y _{0} = x \in l{X} . We assume that l{X} ^{\operatorname{tr} } is almost surely reached, i.e. \forall x \in l{X}, \operatorname{P} _{x(T} < inf ty) = 1 .

The general definition^[1] is: a probability measure

\nu on l{X} ^{a} is said to be a quasi-stationary distribution (QSD) if for every measurable set B contained in l{X} ^{a} , $$\backslash forall\; t\; \backslash geq\; 0,\; \backslash operatorname\_\backslash nu(Y\_t\; \backslash in\; B\; \backslash mid\; T\; >\; t)\; =\; \backslash nu(B)$$where \operatorname{P} _{\nu} = \int ^{a}} _{l{X} \operatorname{P} _{x} d \nu(x) .

In particular

\forall B \in l{B}( l{X} ^{a),} \forall t \geq 0 , \operatorname{P} _{\nu(Y} _{t} \in B, T > t) = \nu(B) \operatorname{P} _{\nu(T} > t).

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:^[2] if

\nu is a QSD then there exists \theta(\nu) > 0 such that \forall t \in N , \operatorname{P} _{\nu(T} > t) = \exp( - \theta(\nu) x t) .

Moreover, for any

\vartheta < \theta(\nu) we get \operatorname{E} ^{\vartheta t} _{\nu(e} ) < inf ty .

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

\theta ^{*} _{x} : = \sup \{ \theta \mid \operatorname{E} ^{\theta T} _{x(e} ) < inf ty \} . A necessary condition for the existence of a QSD is \exists x \in l{X} ^{a,} \theta ^{*} _{x} > 0 and we have the equality \theta ^{*} _{x} = \liminf _{t} - (1)/(t) log (\operatorname{P} _{x(T} > t)).

Moreover, from the previous paragraph, if

\nu is a QSD then \operatorname{E} _{\nu} \left( e ^{\theta(\nu)T} \right) = inf ty . As a consequence, if \vartheta > 0 satisfies \sup ^{a}} _{x \in l{X} \{ \operatorname{E} ^{\vartheta T} _{x(e} ) \} < inf ty then there can be no QSD \nu such that \vartheta = \theta(\nu) because other wise this would lead to the contradiction inf ty = \operatorname{E} _{\nu} \left( e ^{\theta(\nu)T} \right) \leq \sup ^{a}} _{x \in l{X} \{ \operatorname{E} ^{\theta(\nu) T} _{x(e} ) \} < inf ty .(P _{t,} t \geq 0 ) of the process before killing. Then, under the conditions that l{X} ^{a} is a compact Hausdorff space and that P _{1} preserves the set of continuous functions, i.e. P ^{a))} _{1 ( l{C}( l{X}} \subseteq l{C}( l{X} ^{a)} , there exists a QSD.

History

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]

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.

Notes and References

1. Book: Collet. Pierre. Martínez. Servet. San Martín. Jaime. Quasi-Stationary Distributions . en-gb. 10.1007/978-3-642-33131-2. Probability and its Applications. 2013 . 2013. 978-3-642-33130-5.
2. 1427713. Advances in Applied Probability. 24. 4. 795–813. en. Ferrari. Pablo A.. Existence of Non-Trivial Quasi-Stationary Distributions in the Birth-Death Chain. Martínez. Servet. Picco. Pierre. 1992. 10.2307/1427713. 17018407 .
3. WRIGHT, Sewall. Evolution in Mendelian populations. Genetics, 1931, vol. 16, no 2, pp. 97–159.
4. YAGLOM, Akiva M. Certain limit theorems of the theory of branching random processes. In : Doklady Akad. Nauk SSSR (NS). 1947. p. 3.
5. BARTLETT, Mi S. On theoretical models for competitive and predatory biological systems. Biometrika, 1957, vol. 44, no 1/2, pp. 27–42.
6. BARTLETT, Maurice Stevenson. Stochastic population models; in ecology and epidemiology. 1960.
7. VERE-JONES. D.. 1962-01-01. The Quarterly Journal of Mathematics. en. 13. 1. 7–28. 10.1093/qmath/13.1.7. 0033-5606. Geometric Ergodicity in Denumerable Markov Chains. 1962QJMat..13....7V. 10338.dmlcz/102037. free.
8. Darroch. J. N.. Eugene Seneta. Seneta. E.. 1965. On Quasi-Stationary Distributions in Absorbing Discrete-Time Finite Markov Chains. 3211876. Journal of Applied Probability. 2. 1. 88–100. 10.2307/3211876. 67838782 .