An Extreme Value Theory for Sequence Matching

Abstract

Consider finite sequences $X_1, X_2,\cdots, X_m$ and $Y_1, Y_2,\cdots, Y_n$ where the letters ${X_i}$ and ${Y_i}$ are chosen i.i.d. on a countable alphabet with $p=P(X_1=Y_1)\in(0,1)$ We study the distribution of the longest contiguous run of matches between the X's and Y's allowing at most k mismatches. The distribution is closely approximated by that of the maximum of (1 - p)mn i.i.d. negative binomial random variables. The latter distribution is in turn shown to behave like the integer part of an extreme value distribution. The expectation is approximately $\log(qmn)+k\log\log(qmn)+k\log(q/p)-\log(k!)+\gamma\log(e)-\frac{1}{2}$, where q = 1 - p, log denotes logarithm base 1/p, and y is the Euler constant. The variance is approximated by $(\pi\log(e))2/6+\frac{1}{2}$. The paper concludes with an example in which we compare segments taken from the DNA sequence of the bacteriophage lambda.