Maximal Length of Common Words Among Random Letter Sequences

Abstract

Consider random letter sequences {ξ(σ) t, t = 1,..., N; σ = 1,..., s} based on a finite alphabet generated by uniformly mixing stationary processes. The asymptotic distributional properties of the length of the longest common word in r or more of the s sequences Kr,s(N), are investigated. When the probability measures of the different sequences are not too dissimilar, a classical extremal type limit law holds for Kr,s(N) - (r log N/(-log λ)), λ being an appropriate local match parameter. The distributional properties of other long-word relationships and patterns among the sequences are also discussed.