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r-scan statistics of a Poisson process with events transformed by duplications, deletions, and displacements

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  01 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, USA.
∗ Postal address: Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
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Abstract

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A stochastic model of a dynamic marker array in which markers could disappear, duplicate, and move relative to its original position is constructed to reflect on the nature of long DNA sequences. The sequence changes of deletions, duplications, and displacements follow the stochastic rules: (i) the original distribution of the marker array {…, X−2, X−1, X0, X1, X2, …} is a Poisson process on the real line; (ii) each marker is replicated l times; replication or loss of marker points occur independently; (iii) each replicated point is independently and randomly displaced by an amount Y relative to its original position, with the Y displacements sampled from a continuous density g(y). Limiting distributions for the maximal and minimal statistics of the r-scan lengths (collection of distances between r + 1 successive markers) for the l-shift model are derived with the aid of the Chen-Stein method and properties of Poisson processes.

Keywords

r-scan statisticPoisson processChen-Stein methodPoisson approximation

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 3 , September 2007 , pp. 799 - 825
Copyright
Copyright © Applied Probability Trust 2007 

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