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osu1236459876.pdf (363.25 KB)
ETD Abstract Container
Abstract Header
Time to Coalescence for a Class of Nonuniform Allocation Processes
Author Info
McSweeney, John Kingen
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1236459876
Abstract Details
Year and Degree
2009, Doctor of Philosophy, Ohio State University, Mathematics.
Abstract
We study a so-called coalescent process that can be described as follows: start with a set of
n
boxes and
b
0
balls. Let p=(p1,p2,…,pn) be any probability vector. Throw each ball into box
j
with probability
p
j
, independently for each ball. Any balls that land in the same box are fused into a single ball, and the process is repeated with this (possibly smaller) number of balls. Continue this process until there is only one ball left; the time at which this happens is called the coalescence time, denoted
T
. This problem can also be phrased in the context of population genetics, where it is referred to as the
Generalized Wright-Fisher Model
. In that formulation, the balls represent ancestral lineages, and
T
is the the number of generations back in time one has to go to find a common ancestor for b0 individuals from the current generation. We shall mainly study the expected coalescence time
E
[
T
]. For
b
0
=n, and p nonuniform, little is known about the expected time spent when the number of balls is relatively large. We show that for vectors p satisfying a mild uniformity condition, this quantity is negligible compared to the expected time spent when the number of balls is “small”, which is asymptotically 2(p1^2+p2^2+…+pn^2)^(-1). We further show that this condition is sharp, in that if it is not met, there are vectors p which give rise to processes which do not have this qualitative behavior, and thus where the expected coalescence time far exceeds 2(p1^2+p2^2+…+pn^2)^(-1).
Committee
Boris Pittel, PhD (Advisor)
Gerald Edgar, PhD (Committee Member)
Janet Best, PhD (Committee Member)
Pages
94 p.
Subject Headings
Mathematics
Keywords
Stochastic Processes
;
Coalescence
;
Wright-Fisher Model
;
Coupling From the Past
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Citations
McSweeney, J. K. (2009).
Time to Coalescence for a Class of Nonuniform Allocation Processes
[Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1236459876
APA Style (7th edition)
McSweeney, John.
Time to Coalescence for a Class of Nonuniform Allocation Processes.
2009. Ohio State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1236459876.
MLA Style (8th edition)
McSweeney, John. "Time to Coalescence for a Class of Nonuniform Allocation Processes." Doctoral dissertation, Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1236459876
Chicago Manual of Style (17th edition)
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Document number:
osu1236459876
Download Count:
560
Copyright Info
© 2009, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.