Large deviation results for a class of Markov chains with application to an infinite alleles model of population genetics (Q1203749)

Let \(\{X_ n^{(N)}\}\) be a Markov chain in the probability measures \(P[0,1]\). Define the exit time \(T=\inf\{n: X_ n\neq D\}\) for some open ball \(D\) once \(X_ 0\in D\). Assuming various regularity conditions it is shown that the expected time \(T\) is logarithmically equivalent to \(\exp\{M(N)V\}\) for some \(M(N)\to \infty\), \(V > 0\). These results apply to an infinite alleles model in population genetics. The present paper is in turn an extension of the work of \textit{G. J. Morrow} and \textit{S. Sawyer} [see Ann. Probab. 17, No. 3, 1124-1146 (1989; Zbl 0684.60018)], where an \(R^ d\)-valued Markov chain was considered.