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

Part of: Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Samuel Karlin  and
Amir Dembo
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.
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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.

Keywords

MAXIMAL SEGMENTAL SUMSMARKOV-DEPENDENT WALD MARTINGALESMARKOV (WIENER-HOPF) THEORY

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G70: Extreme value theory; extremal processes 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 113 - 140
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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