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Time to Coalescence for a Class of Nonuniform Allocation Processes
McSweeney, John Kingen

2009, Doctor of Philosophy, Ohio State University, Mathematics.
We study a so-called coalescent process that can be described as follows: start with a set of n boxes and b0 balls. Let p=(p1,p2,…,pn) be any probability vector. Throw each ball into box j with probability pj, 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 b0=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).
Boris Pittel, PhD (Advisor)
Gerald Edgar, PhD (Committee Member)
Janet Best, PhD (Committee Member)
94 p.

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McSweeney, J. (2009). Time to Coalescence for a Class of Nonuniform Allocation Processes. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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McSweeney, John. "Time to Coalescence for a Class of Nonuniform Allocation Processes." Electronic Thesis or Dissertation. Ohio State University, 2009. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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McSweeney, John "Time to Coalescence for a Class of Nonuniform Allocation Processes." Electronic Thesis or Dissertation. Ohio State University, 2009. https://etd.ohiolink.edu/

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