  Given two sequences over a finite alphabet $\mathcal{L}$, the $D_2$ 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 $D_2$ statistic in the context
of DNA sequences, under the assumption of strand symmetric Bernoulli text. For
$k<m$, 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.
