Publication Date
2015-07-31
Availability
Open access
Embargo Period
2015-07-31
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PHD)
Department
Mathematics (Arts and Sciences)
Date of Defense
2015-06-30
First Committee Member
Ilie Grigorescu
Second Committee Member
George C. Cosner
Third Committee Member
Victor C. Pestien
Fourth Committee Member
Burton J. Rosenberg
Abstract
We derive a fluid limit for a multi-type urn model, also known as a hydrodynamic limit, in the sense that random trajectories of the microscopic process are shown to follow, as the scaling factor L, proportional with the initial population, approaches infinity, a unique trajectory characterized as the strong global solution of a specific dynamical system (the macroscopic equation). The result is a weak Law of Large Numbers of random variables with values in the Skorokhod space of right continuous with left limits functions with values in R^k , where k is the number of types. We obtain that while the macroscopic process does not vanish in finite time, the microscopic process has a probability of extinction no larger than O(1/L) . A similar limit is proven for the normalized vector of population proportions, together with qualitative results on its asymptotic behavior. Both limits are in probability, uniformly in time for any fixed time interval. The model and the scaling studied is inspired by earlier work by Schreiber, Benaïm et al. and generalizes the well known Replicator model with applications in mathematical ecology and genome population dynamics.
Keywords
hydrodynamic limit; scaling; urn model; replicator model
Recommended Citation
Zhang, Zhe, "Scaling Limit of a Generalized Pólya Urn Model" (2015). Open Access Dissertations. 1490.
https://scholarlyrepository.miami.edu/oa_dissertations/1490