Maximal Length of Common Words Among Random Letter Sequences

Abstract

Consider random letter sequences $\{\xi^{(\sigma)}_t, t = 1,\ldots, N; \sigma = 1,\ldots, s\}$ 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 $K_{r,s}(N)$, are investigated. When the probability measures of the different sequences are not too dissimilar, a classical extremal type limit law holds for $K_{r,s}(N) - (r \log N/(-\log \lambda)), \lambda$ being an appropriate local match parameter. The distributional properties of other long-word relationships and patterns among the sequences are also discussed.