On the Order of the Central Moments of the Length of the Longest Common Subsequences in Random Words

Abstract

Abstract. We investigate the order of the r-th, 1 ≤ r < +∞, central moment of the length of the longest common subsequence of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, a lower bound is shown to be of order n r/2 . This result complements a generic upper bound also of order n r/2 .

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

“…In many ways, the present paper complements [21]. The results of [19] (the so-called low entropy case) were generalized in [16], and a linear variance lower bound was proved for an arbitrary finite alphabet (not just binary) and for the central moments of arbitrary order (not just the variance), but still under a strongly asymmetric distribution over the letters. The goal of the present paper is to study in this general framework the lower bounds on generalized moments of the optimal alignment score L n .…”

“…random sequences such that Var(L n ) admits a sublinear (in n) lower bound. The result in [15] upper-bounds the Monge-Kantorovich-Wasserstein distance, and in turn the Kolmogorov distance, implying weak convergence. The limiting theorem in [15] was extended to multiple independent i.i.d.…”

“…The result in [15] upper-bounds the Monge-Kantorovich-Wasserstein distance, and in turn the Kolmogorov distance, implying weak convergence. The limiting theorem in [15] was extended to multiple independent i.i.d. sequences with a general score function in [12].…”

“…This observation suggests to conjecture that (1.3) also holds in more general cases, in particular, in the LCS case. No such type of limiting theorem was known until [15], which proved (1.3) in the LCS case, when X n and Y n are independent i.i.d. random sequences such that Var(L n ) admits a sublinear (in n) lower bound.…”

“…sequences with a general score function in [12]. In contrast to [15], the result in [12] directly upper-bounds the Kolmogorov distance, which improves the rate of weak convergence, but still requires a (looser) sublinear variance lower bound. Hence, establishing a linear-order variance lower bound for a general distribution of X 1 implies immediately the corresponding limiting theorem (1.3).…”