First occurrence in pairs of long words: a Penney-ante conjecture of Pevzner

Abstract

Suppose X1,X2,⋯ is a sequence of independent and identically distributed random elements taking values in a finite set S of size |S|≥2 with probability distribution P(X=s)=p(s)>0 for s∈S. P. Pevzner [Kvantl 5 (1987), 4--15; per bibl.] has conjectured that for every probability distribution P there exists an N>0 such that for every word A with letters in S whose length is at least N, there exists a second word B of the same length as A, such that the event that B appears before A in the sequence X1,X2,⋯ has greater probability than that of A appearing before B. In this paper it is shown that a distribution P satisfies Pevzner's conclusion if and only if the maximum value of P, p, and the secondary maximum c satisfy the inequality c>p(1−p)/(1+p). For |S|=2 or |S|=3, the inequality is true and the conjecture holds. If c≤p(1−p)/(1+p), then the conjecture is true when A is not allowed to consist of pure repetitions of that unique element for which the distribution takes on its mode.