## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 3 (1989), 1124-1146.

### Large Deviation Results for a Class of Markov Chains Arising from Population Genetics

Gregory J. Morrow and Stanley Sawyer

#### Abstract

Let $\{X_n\}$ be a Markov chain on a bounded set in $R^d$ with $E_x(X_1) = f_N(x) = x + \beta_N h_N(x)$, where $x_0$ is a stable fixed point of $f_N(x) = x$, and $\operatorname{Cov}_x(X_1) \approx \sigma^2(x)/N$ in various senses. Let $D$ be an open set containing $x_0$, and assume $h_N(x) \rightarrow h(x)$ uniformly in $D$ and either $\beta_N \equiv 1$ or $\beta_N \rightarrow 0, \beta_N \gg \sqrt{\log N/N}$. Then, assuming various regularity conditions and $X_0 \in D$, the time the process takes to exit from $D$ is logarithmically equivalent in probability to $e^{VN\beta_N}$, where $V > 0$ is the solution of a variational problem of Freidlin-Wentzell type $\lbrack \text{if} \beta_N \rightarrow 0 \text{and} d = 1, V = \inf\{2 \int^y_{x_0}\sigma^{-2}(u)|h(u) du|: y \in \partial D\} \rbrack$. These results apply to the Wright-Fisher model in population genetics, where $\{X_n\}$ represent gene frequencies and the average effect of forces such as selection and mutation are much stronger than effects due to finite population size.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 3 (1989), 1124-1146.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991260

**Digital Object Identifier**

doi:10.1214/aop/1176991260

**Mathematical Reviews number (MathSciNet)**

MR1009448

**Zentralblatt MATH identifier**

0684.60018

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 92A10 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Large deviations Markov chains Ventsel-Freidlin Wright-Fisher population genetics

#### Citation

Morrow, Gregory J.; Sawyer, Stanley. Large Deviation Results for a Class of Markov Chains Arising from Population Genetics. Ann. Probab. 17 (1989), no. 3, 1124--1146. doi:10.1214/aop/1176991260. https://projecteuclid.org/euclid.aop/1176991260