Josh Sparks Friday March 1st 2013

Title: Pólya Urn Scheme in Continuous TJosh_Sparksime


Josh Sparks presents a piece of his master’s dissertation discussing Polya-Urn Models. This seminar should be of particular interest to those currently studying Polya-Urns in Stat6289 with Professor Mahmoud.


Josh Sparks is an accomplished academic. He graduated at the top of his class from Eastern Kentucky University with a double bachelors in arts and science. During his tenure at EKU he quadruple majored in mathematics, statistics, sociology and and cultural studies. Josh subsequently went on to obtain a master’s in Sociology from the University of Toronto and is currently completing his master’s in statistics at The George Washington University.


The Pólya process is obtained by embedding the usual (discrete-time) Pólya urn scheme in continuous time. We study the class of tenable Pólya processes of white and blue balls with zero balance (no change in n, the total number of balls, over time). This class includes the (continuous-time) Ehrenfest process and the (continuous-time) Coupon Collector’s process. We look at the composition of the urn at time t_n (dependent on n). We identify a critical phase of t_n at the edges of which phase transitions occur. In the subcritical phase, under proper scaling the number of white balls is concentrated around a constant. In the critical phase, we have sufficient variability for an asymptotic normal distribution to be in effect. In this phase, the influence of the initial conditions is still somewhat pronounced. Beyond the critical phase, the urn is very well mixed with an asymptotic normal distribution, in which all initial conditions wither away. The results are obtained by an analytic approach utilizing partial differential equations.


Date: FridayMarch 1st 2013

Time: 4-5 pm

Location: MON 251 ( MONROE HALL 2115 G St, NW  for map click:

Audience: All