Large Deviation Results for a Class of Markov Chains with Application to an Infinite Alleles Model of Population Genetics

Abstract

Let $\{X_n\} = \{X^{(N)}_n\}$ be a Markov chain in the probability measures $\mathbf{P}\lbrack 0, 1\rbrack$, equipped with a certain metric for the topology of weak convergence, and denote $\mathbf{E}_x(X_1) = \mathbf{f}_N(x)$. Define a projection $\xi = \pi_dx$ on $\mathbf{P}\lbrack 0, 1\rbrack$ by defining $\xi$ to be absolutely continuous with respect to Lebesgue measure with constant density $(d + 1)x(A)$ on each interval $A$ of an equipartition of $\lbrack 0, 1 \rbrack$ into $d + 1$ intervals, and let $\pi_d\mathbf{P}\lbrack 0, 1\rbrack \subset \mathbb{R}^d$ be the natural embedding. Assume $\pi_d\mathbf{f}_N(\xi) \underline{\sim} \xi + \beta_N h_d(\xi), h_d(\xi_{0, d}) = \mathbf{0}$ and $\operatorname{Cov}_\xi(\pi_d X_1) \underline{\sim} \sigma^2_d(\xi)/N$, in certain senses as $N \rightarrow \infty$, where $\xi_{0, d}$ is an asymptotically stable fixed point of $\pi_d\mathbf{f}_N(\xi) = \xi$ and $x_0 = \lim_{d \rightarrow \infty} \xi_{0, d}$ exists in $\mathbf{P}\lbrack 0, 1\rbrack$. Assuming various regularity conditions and $\beta = \beta_N \rightarrow 0, N\beta/\log N \rightarrow \infty$, it is shown that the expected time it takes the Markov chain to exit a fixed open ball $D$ about $x_0$ once $X_0 \in D$ is logarithmically equivalent to $\exp\lbrack N\beta_N V \rbrack$, where $V > 0$ is a limit of solutions $V_d = V_d(h_d, \sigma_d)$ of variational problems of Wentzell-Freidlin type in $\mathbb{R}^d$ as $d \rightarrow \infty$. These results apply to an infinite alleles model in population genetics, where $\{X_n\}$ represents the evolution of distributions of types among a population of $N$ randomly mating genes, and where forces of mutation and selection are stronger than effects due to finite population size.