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Large deviation results for a class of Markov chains with application to an infinite alleles model of population genetics. (English) Zbl 0767.92019

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 G. J. Morrow and S. Sawyer [see Ann. Probab. 17, No. 3, 1124-1146 (1989; Zbl 0684.60018)], where an \(R^ d\)-valued Markov chain was considered.

MSC:

92D10 Genetics and epigenetics
60F10 Large deviations
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J05 Discrete-time Markov processes on general state spaces

Citations:

Zbl 0684.60018
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