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

Let {Xn}={X(N)n} be a Markov chain in the probability measures P[0,1], equipped with a certain metric for the topology of weak convergence, and denote Ex(X1)=fN(x). Define a projection .=.dx on P[0,1] by defining . to be absolutely continuous with respect to Lebesgue measure with constant density (d+1)x(A) on each interval A of an equipartition of [0,1] into d+1 intervals, and let .dP[0,1].Rd be the natural embedding. Assume .dfN(.)...+.Nhd(.),hd(.0,d)=0 and Cov.(.dX1)...2d(.)/N, in certain senses as N.., where .0,d is an asymptotically stable fixed point of .dfN(.)=. and x0=limd...0,d exists in P[0,1]. Assuming various regularity conditions and .=.N.0,N./logN.., it is shown that the expected time it takes the Markov chain to exit a fixed open ball D about x0 once X0.D is logarithmically equivalent to exp[N.NV], where V>0 is a limit of solutions Vd=Vd(hd,.d) of variational problems of Wentzell-Freidlin type in Rd as d... These results apply to an infinite alleles model in population genetics, where {Xn} 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.