Quasi-stationary distribution

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

The general definition ^[1] is: a probability measure ${\displaystyle \nu }$ on ${\displaystyle {\mathcal {X}}^{a}}$ is said to be a quasi-stationary distribution (QSD) if for every measurable set ${\displaystyle B}$ contained in ${\displaystyle {\mathcal {X}}^{a}}$,

$${\displaystyle \forall t\geq 0,\operatorname {P} _{\nu }(Y_{t}\in B\mid T>t)=\nu (B)}$$

where ${\displaystyle \operatorname {P} _{\nu }=\int _{{\mathcal {X}}^{a}}\operatorname {P} _{x}\,\mathrm {d} \nu (x)}$.

In particular ${\displaystyle \forall B\in {\mathcal {B}}({\mathcal {X}}^{a}),\forall t\geq 0,\operatorname {P} _{\nu }(Y_{t}\in B,T>t)=\nu (B)\operatorname {P} _{\nu }(T>t).}$

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: ^{[1] [2]} if ${\displaystyle \nu }$ is a QSD then there exists ${\displaystyle \theta (\nu )>0}$ such that ${\displaystyle \forall t\in \mathbf {N} ,\operatorname {P} _{\nu }(T>t)=\exp(-\theta (\nu )\times t)}$.

Moreover, for any ${\displaystyle \vartheta <\theta (\nu )}$ we get ${\displaystyle \operatorname {E} _{\nu }(e^{\vartheta t})<\infty }$.

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

Moreover, from the previous paragraph, if ${\displaystyle \nu }$ is a QSD then ${\displaystyle \operatorname {E} _{\nu }\left(e^{\theta (\nu )T}\right)=\infty }$. As a consequence, if ${\displaystyle \vartheta >0}$ satisfies ${\displaystyle \sup _{x\in {\mathcal {X}}^{a}}\{\operatorname {E} _{x}(e^{\vartheta T})\}<\infty }$ then there can be no QSD ${\displaystyle \nu }$ such that ${\displaystyle \vartheta =\theta (\nu )}$ because other wise this would lead to the contradiction ${\displaystyle \infty =\operatorname {E} _{\nu }\left(e^{\theta (\nu )T}\right)\leq \sup _{x\in {\mathcal {X}}^{a}}\{\operatorname {E} _{x}(e^{\theta (\nu )T})\}<\infty }$.

A sufficient condition for a QSD to exist is given considering the transition semigroup ${\displaystyle (P_{t},t\geq 0)}$ of the process before killing. Then, under the conditions that ${\displaystyle {\mathcal {X}}^{a}}$ is a compact Hausdorff space and that ${\displaystyle P_{1}}$ preserves the set of continuous functions, i.e. ${\displaystyle P_{1}({\mathcal {C}}({\mathcal {X}}^{a}))\subseteq {\mathcal {C}}({\mathcal {X}}^{a})}$, 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.

References

1. Collet, Pierre; Martínez, Servet; San Martín, Jaime (2013). Quasi-Stationary Distributions | SpringerLink. Probability and its Applications. doi:10.1007/978-3-642-33131-2. ISBN   978-3-642-33130-5.
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.