We consider local alignments without gaps of two independent Markov chains from a finite alphabet, and we derive sufficient conditions for the number of essentially different local alignments with a score exceeding a high threshold to be asymptotically Poisson distributed. From the Poisson approximation a Gumbel approximation of the maximal local alignment score is obtained. The results extend those obtained by Dembo, Karlin and Zeitouni [Ann. Probab. 22 (1994) 2022–2039] for independent sequences of i.i.d. variables.
"Local alignment of Markov chains." Ann. Appl. Probab. 16 (3) 1262 - 1296, August 2006. https://doi.org/10.1214/105051606000000321