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Counts of long aligned word matches among random letter sequences

Published online by Cambridge University Press:  01 July 2016

Samuel Karlin  and
Friedemann Ost
Samuel Karlin*
Affiliation:
Stanford University
Friedemann Ost*
Affiliation:
Technische Universität München
*
∗Postal address: Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
∗∗ Postal address: Institut für Angewandte Mathematik und Statistik, Technische Universität München, Arcisstr. 21, D-8000 München 2, Germany.
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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.

Keywords

LONGEST SUCCESS RUNLOCAL MATCH DISTRIBUTIONSCHUR PRODUCT MATRIX
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 2 , June 1987 , pp. 293 - 351
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

Research supported in part by NIH Grant GM10452-22 and NSF Grant MCS82-15131.

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