Standard deviation of the longest common subsequence

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.