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Quasi-stationary distribution

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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

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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[1] is: a probability measure on is said to be a quasi-stationary distribution (QSD) if for every measurable set contained in , where .

In particular

General results

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Killing time

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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 such that .

Moreover, for any we get .

Existence of a quasi-stationary distribution

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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

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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

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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.

References

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  1. ^ a b Collet, Pierre; Martínez, Servet; San Martín, Jaime (2013). Quasi-Stationary Distributions. Probability and its Applications. doi:10.1007/978-3-642-33131-2. ISBN 978-3-642-33130-5.{{cite book}}: CS1 maint: date and year (link)
  2. ^ Ferrari, Pablo A.; Martínez, Servet; Picco, Pierre (1992). "Existence of Non-Trivial Quasi-Stationary Distributions in the Birth-Death Chain". Advances in Applied Probability. 24 (4): 795–813. doi:10.2307/1427713. JSTOR 1427713. S2CID 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). "Geometric Ergodicity in Denumerable Markov Chains". The Quarterly Journal of Mathematics. 13 (1): 7–28. Bibcode:1962QJMat..13....7V. doi:10.1093/qmath/13.1.7. hdl:10338.dmlcz/102037. ISSN 0033-5606.
  8. ^ Darroch, J. N.; Seneta, E. (1965). "On Quasi-Stationary Distributions in Absorbing Discrete-Time Finite Markov Chains". Journal of Applied Probability. 2 (1): 88–100. doi:10.2307/3211876. JSTOR 3211876. S2CID 67838782.