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

TL;DR

This paper proves a central limit theorem for the distribution of the optimal alignment score among multiple independent random sequences over a finite alphabet, under certain regularity conditions.

Contribution

It establishes a CLT for the optimal alignment score of multiple random words, extending previous results to a broader setting with fewer restrictions.

Findings

CLT holds for the optimal alignment score under variance lower-bound

Score distribution converges to a normal distribution

Applicable to permutation-invariant, bounded score functions

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.