  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.
