We study approximations to the distribution of counts of matches in the best matching segment of specified length when comparing two long sequences of i.i.d. letters. The key tools used are large-deviation inequalities and the Chen-Stein method of Poisson approximation. The origin of the problem in molecular biology is indicated.
"The Erdos-Renyi Law in Distribution, for Coin Tossing and Sequence Matching." Ann. Statist. 18 (2) 539 - 570, June, 1990. https://doi.org/10.1214/aos/1176347615