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Limit distributions of maximal segmental score among Markov-dependent partial sums

Published online by Cambridge University Press:  01 July 2016

Samuel Karlin*
Affiliation:
Stanford University
Amir Dembo*
Affiliation:
Stanford University
*
Postal address: Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
∗∗Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, USA.

Abstract

Let s1, …, sn be generated governed by an r-state irreducible aperiodic Markov chain. The partial sum process is determined by a realization of states with s0 = α and the real-valued i.i.d. bounded variables Xαß associated with the transitions si = α, si+1 = β. Assume Χ αβ has negative stationary mean. The explicit limit distribution of the maximal segmental sum is derived. Computational methods with potential applications to the analysis of random Markov-dependent letter sequences (e.g. DNA and protein sequences) are presented.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

Research partly supported by NIH Grants GM39907–02, GM10452–27, and NSF Grant DMS86–06244.

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