## 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

Zbl 0684.60018
Full Text: