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Poisson approximations for conditional r-scan lengths of multiple renewal processes and application to marker arrays in biomolecular sequences

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Chingfer Chen  and
Samuel Karlin
Chingfer Chen*
Affiliation:
Stanford University
Samuel Karlin*
Affiliation:
Stanford University
*
∗Postal address: Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA
∗∗Email address: fd.zgg@forsythe.stanford.edu
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Abstract

This study is motivated by problems of molecular sequence comparison for multiple marker arrays with correlated distributions. In this paper, the model assumes two (or more) kinds of markers, say Markers A and B, distributed along the DNA sequence. The two primary conditions of interest are (i) many of Marker B (say ≥ m) occur, and (ii) few of Marker B (say ≤ l) occur. We title these the conditional r-scan models, and inquire on the extent to which Marker A clusters or is over-dispersed in regions satisfying condition (i) or (ii). Limiting distributions for the extremal r-scan statistics from the A array satisfying conditions (i) and (ii) are derived by extending the Chen-Stein Poisson approximation method.

Keywords

r-scan statisticsChen-Stein methodtotal variation distancetotal positivity

MSC classification

Primary: 60E05: Distributions
Secondary: 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 865 - 880
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

Supported in part by NIH Grant 2R01HG00335-11 and 5R01GM10452-35.

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