Asymptotic behavior of k-word matches between two uniformly distributed sequences

Given two sequences of length n over a finite alphabet A of size \A\ = d, the D₂ statistic is the number of k-letter word matches between the two sequences. This statistic is used in bioinformatics for EST sequence database searches. Under the assumption of independent and identically distributed letters in the sequences, Lippert, Huang and Waterman (2002) raised questions about the asymptotic behavior of D₂ when the alphabet is uniformly distributed. They expressed a concern that the commonly assumed normality may create errors in estimating significance. In this paper we answer those questions. Using Stein's method, we show that, for large enough k, the D₂ statistic is approximately normal as n gets large. When k = 1, we prove that, for large enough d, the D₂ statistic is approximately normal as n gets large. We also give a formula for the variance of D₂ in the uniform case.