Alea 3, 1–29 (2007) Deviation from mean in sequence comparison with a periodic sequence Heinrich Matzinger, J¨ uri Lember and Clement Durringer Heinrich Matzinger, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany Georgia Tech School of Mathematics, Atlanta, Georgia 30332-0160, U.S.A. E-mail address : matzing@math.gatech.edu J¨ uri Lember, Tartu University, Institute of Mathematical Statistics, Liivi 2-513 50409, Tartu, Estonia E-mail address : jyril@ut.ee Clement Durringer, University of Toulouse, 5 all´ee Antonio Machado, 31058 Toulouse Cedex 9, France E-mail address : durringe@cict.fr Abstract. Let L n denote the length of the longest common subsequence of two sequences of length n. We draw one of the sequences i.i.d., but the other is non- random and periodic. We prove that VAR[L n ] = Θ(n). For such setup, our result rejects the Chvatal-Sankoff conjecture (1975) that VAR[L n ]= o(n 2 3 ) and answers to Waterman’s question (1994), whether the linear bound on VAR[L n ] can be im- proved. 1. Introduction Let {X i } i∈N and {Y i } i∈N be two ergodic processes independent of each other. We assume that the variables X i and Y i have a common state space. Let X := X 1 X 2 ...X n and Y := Y 1 Y 2 ...Y n . A common subsequence of X and Y is a sub- sequence that is contained in X and in Y . Formally, a common subsequence of X and Y consists of two subsets of indices {i 1 ,...,i k }, {j 1 ,...,j k }⊂{1,...,n} such that X i1 = Y i1 ,X i2 = Y i2 ,...,X i k = Y i k . The length of such a common subsequence is k. The longest common subsequence (LCS) of X and Y is any common subsequence that has the longest possible length, denoted by L n . The random variable L n is the main object of the paper. Received by the editors November 30, 2005 ; accepted December 4, 2006. 2000 Mathematics Subject Classification. 60K35, 41A25, 60C05. Key words and phrases. Longest common subsequence, variance bound, Chvatal-Sankoff conjecture. J¨ uri Lember is supported by Estonian Science Foundation Grant nr. 5694. 1