The Annals of Probability

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

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

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