Time to Coalescence for a Class of Nonuniform Allocation Processes
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2009
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ndltd-OhioLink-oai-etd.ohiolink.edu-osu12364598762021-08-03T05:55:20Z Time to Coalescence for a Class of Nonuniform Allocation Processes McSweeney, John Kingen Mathematics Stochastic Processes Coalescence Wright-Fisher Model Coupling From the Past We study a so-called coalescent process that can be described as follows: start with a set of <i>n</i> boxes and <i>b</i><sub>0</sub> balls. Let p=(p1,p2,…,pn) be any probability vector. Throw each ball into box <i>j</i> with probability <i>p</i><sub>j</sub>, 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 <i>T</i>. This problem can also be phrased in the context of population genetics, where it is referred to as the <i>Generalized Wright-Fisher Model</i>. In that formulation, the balls represent ancestral lineages, and <i>T</i> 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 <i>E</i>[<i>T</i>]. For <i>b</i><sub>0</sub>=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). 2009-08-27 English text The Ohio State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=osu1236459876 http://rave.ohiolink.edu/etdc/view?acc_num=osu1236459876 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
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NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics Stochastic Processes Coalescence Wright-Fisher Model Coupling From the Past |
| spellingShingle |
Mathematics Stochastic Processes Coalescence Wright-Fisher Model Coupling From the Past McSweeney, John Kingen Time to Coalescence for a Class of Nonuniform Allocation Processes |
| author |
McSweeney, John Kingen |
| author_facet |
McSweeney, John Kingen |
| author_sort |
McSweeney, John Kingen |
| title |
Time to Coalescence for a Class of Nonuniform Allocation Processes |
| title_short |
Time to Coalescence for a Class of Nonuniform Allocation Processes |
| title_full |
Time to Coalescence for a Class of Nonuniform Allocation Processes |
| title_fullStr |
Time to Coalescence for a Class of Nonuniform Allocation Processes |
| title_full_unstemmed |
Time to Coalescence for a Class of Nonuniform Allocation Processes |
| title_sort |
time to coalescence for a class of nonuniform allocation processes |
| publisher |
The Ohio State University / OhioLINK |
| publishDate |
2009 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=osu1236459876 |
| work_keys_str_mv |
AT mcsweeneyjohnkingen timetocoalescenceforaclassofnonuniformallocationprocesses |
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1719427843393323008 |