A Limit Theorem for Matching Random Sequences Allowing Deletions

Abstract

We consider a sequence matching problem involving the optimal alignment score for contiguous sequences, rewarding matches by one unit and penalizing for deletions and mismatches by parameters $\delta$ and $\mu$, respectively. Let $M_n$ be the optimal score over all possible choices of two contiguous regions. Arratia and Waterman conjectured that, when the score constant $a(\mu, \delta) < 0$, $P\big(\frac{M_n}{\log n} \rightarrow 2b\big) = 1$ for some constant $b$. Here we prove the conjecture affirmatively.