Author
McSweeney, John

Year
2009

Advisor
Pittel, Boris

Abstract
We study a so-called coalescent process that can be described as follows:
start with a set of n boxes and b 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 b individuals from the current generation. We shall mainly
study the expected coalescence time E[T]. For b=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 $\mbp$ 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+...+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+...+pn^2)^(-1).

Thesis
McSweeneyThesisMar7.pdf