A Central Limit Theorem for the Optimal Alignments Score in Multiple Random Words

Abstract: Let $\mathbf{X}^{(1)}{n},\ldots,\mathbf{X}^{(m)}{n}$, where $\mathbf{X}^{(i)}{n}=(X^{(i)}{1},\ldots,X^{(i)}_{n})$, $i=1,\ldots,m$, be $m$ independent sequences of independent and identically distributed random variables taking their values in a finite alphabet $\mathcal{A}$. Let the score function $S$, defined on $\mathcal{A}^{m}$, be non-negative, bounded, permutation-invariant, and satisfy a bounded differences condition. Under a variance lower-bound assumption, a central limit theorem is proved for the optimal alignments score of the $m$ random words.