  We present a general approach to the problem of determining tight asymptotic
lower bounds for generalized central moments of the optimal alignment score of
two independent sequences of i.i.d. random variables. At first, these are
obtained under a main assumption for which sufficient conditions are provided.
When the main assumption fails, we nevertheless develop a "uniform
approximation" method leading to asymptotic lower bounds. Our general results
are then applied to the length of the longest common subsequence of binary
strings, in which case asymptotic lower bounds are obtained for the moments and
the exponential moments of the optimal score. As a byproduct, a local upper
bound on the rate function associated with the length of the longest common
subsequences of two binary strings is also obtained.
