  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.
