Standard deviation of the longest common subsequence

TL;DR

This paper proves that the standard deviation of the longest common subsequence length of two Bernoulli sequences grows proportionally to the square root of sequence length when the Bernoulli parameter is small, confirming Waterman's conjecture.

Contribution

It establishes the order of the standard deviation of L_n as √n for small Bernoulli parameters, validating a specific conjecture in sequence alignment theory.

Findings

Standard deviation of L_n is proportional to √n for small Bernoulli parameters.

Validates Waterman's conjecture on the order of standard deviation.

Contradicts the order conjectured by Chvatal and Sankoff.

Abstract

Let $L_n$ be the length of the longest common subsequence of two independent i.i.d. sequences of Bernoulli variables of length $n$. We prove that the order of the standard deviation of $L_n$ is $\sqrt{n}$, provided the parameter of the Bernoulli variables is small enough. This validates Waterman's conjecture in this situation [Philos. Trans. R. Soc. Lond. Ser. B 344 (1994) 383--390]. The order conjectured by Chvatal and Sankoff [J. Appl. Probab. 12 (1975) 306--315], however, is different.