Approximate word matches between two random sequences

Given two sequences over a finite alphabet L, the D₂ statistic is the number of m-letter word matches between the two sequences. This statistic is used in bioinformatics for expressed sequence tag database searches. Here we study a generalization of the D₂ statistic in the context of DNA sequences, under the assumption of strand symmetric Bernoulli text. Fork < m, we look at the count of m-letter word matches with up to k mismatches. For this statistic, we compute the expectation, give upper and lower bounds for the variance and prove its distribution is asymptotically normal.