Lower Bounds on the Generalized Central Moments of the Optimal Alignments Score of Random Sequences

Abstract

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.

A general method for lower bounds on fluctuations of random variables

“…Some variants of this coupling approach for lower bounds on fluctuations are already present in [24,29,30,35,55] for the specific problems handled in those papers; but the potential generality of the idea was not recognized in earlier works. Janson [31] proved a similar lemma, but where Y was assumed to have the same law as X.…”

“…The main lemma of [31] is closely related to the method of this paper. (4) Problem-specific techniques, as in [24,29,30,35,55]. Some of these are also related to the method proposed here.…”

Anti-concentration of random variables from zero-free regions

“…Another route towards anti-concentration is by a coupling approach: when the variable of interest is a function of a random environment, one can often couple two instances of the environment so that one instance of the variable is larger than the other. This approach was taken by Wehr and Aizenman [36] to yield lower bounds on certain variances in the context of the Ising model (and other related models) and is also taken up in other ad-hoc approaches to proving lower bounds on fluctuations [4,14,16,17,18,20,32] which culminated in a recent unifying work of Chatterjee [6].…”

Lower bounds for fluctuations in first-passage percolation for general distributions

“…In contrast to [29] and [11], our methods do not rely on specific properties of the underlying distribution, and explore a more combinatorial avenue. The main idea is inspired by the study of the longest common subsequence problem (see [17,19,25]), and involves introducing a fluctuation in the number of hi-mode weights (weights in the top part of the distribution of t e ). The notion of hi-mode weights was introduced and used in the Ph.D. thesis of Xu [32], where lower bounds of order n r(1−α) 2 are obtained for the r-th central moments (r ≥ 1) in a related last-passage percolation model over an n × ⌊n α ⌋ grid.…”