Distributional regimes for the number of k-word matches between two random sequences

When comparing two sequences, a natural approach is to count the number of k-letter words the two sequences have in common. No positional information is used in the count, but it has the virtue that the comparison time is linear with sequence length. For this reason this statistic D(2) and certain transformations of D(2) are used for EST sequence database searches. In this paper we begin the rigorous study of the statistical distribution of D(2). Using an independence model of DNA sequences, we derive limiting distributions by means of the Stein and Chen-Stein methods and identify three asymptotic regimes, including compound Poisson and normal. The compound Poisson distribution arises when the word size k is large and word matches are rare. The normal distribution arises when the word size is small and matches are common. Explicit expressions for what is meant by large and small word sizes are given in the paper. However, when word size is small and the letters are uniformly distributed, the anticipated limiting normal distribution does not always occur. In this situation the uniform distribution provides the exception to other letter distributions. Therefore a naive, one distribution fits all, approach to D(2) statistics could easily create serious errors in estimating significance.