Open Access
July, 1989 Large Deviation Results for a Class of Markov Chains Arising from Population Genetics
Gregory J. Morrow, Stanley Sawyer
Ann. Probab. 17(3): 1124-1146 (July, 1989). DOI: 10.1214/aop/1176991260

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.

Citation

Download Citation

Gregory J. Morrow. Stanley Sawyer. "Large Deviation Results for a Class of Markov Chains Arising from Population Genetics." Ann. Probab. 17 (3) 1124 - 1146, July, 1989. https://doi.org/10.1214/aop/1176991260

Information

Published: July, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0684.60018
MathSciNet: MR1009448
Digital Object Identifier: 10.1214/aop/1176991260

Subjects:
Primary: 60F10
Secondary: 60G40 , 60J10 , 92A10

Keywords: large deviations , Markov chains , Population genetics , Ventsel-Freidlin , Wright-Fisher

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • July, 1989
Back to Top