Counts of long aligned word matches among random letter sequences

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Abstract
Asymptotic distributional properties of the maximal length aligned word (a contiguous set of letters) among multiple random Markov dependent sequences composed of letters from a finite alphabet are given. For sequences of length N, Cr,s(N) defined as the longest common aligned word found in r or more of s sequences has order growth log N/(–logλ) where λis the maximal eigenvalue of r-Schur product matrices from among the collections of Markov matrices that generate the sequences. The count Z∗r,s(N, k) of positions that initiate an aligned match of length exceeding k = log N/(–logλ) + x but fail to match at the immediately preceding position has a limiting Poisson distribution. Distributional properties of other long aligned word relationships and patterns are also discussed.

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