Chris graduated from the University of Bath in 2019 with an MMath, where he picked up an interest in both measure theory and graph theory. For his final year project, he reviewed works on Lévy processes and continuous-state branching processes with focus on the limiting population of continuous-state branching processes. He wishes to pursue his interest in probability theory, and he is excited to learn about stochastic differential equations. In his spare time he enjoys poker, playing the guitar and he is learning to bake (somewhat successfully).

Research project title:
Limit Theorems for generalised urn schemes

Supervisor(s):
Cécile Mailler, Mathew Penrose

Project description:
Generalised urn schemes are a classical discrete-time probabilistic model that describe the contents of an urn that contains balls of different colours. At each time step, a ball is chosen uniformly at random in the urn, and replaced into the urn together with a set of new balls whose number and colours depend on the colour of the selected ball and on a replacement rule, which is encoded in a matrix R. The cases of R being either the identity matrix or irreducible are well-studied in the literature and limiting theorems show how the composition of the urn behaves when time goes to infinity. In the case of the identity matrix, recent developments by Borovkov have been made on the convergence of the composition of the urn as the number of initial balls goes to infinity together with time; different behaviours have been observed, depending on the scaling of these two factors. Chris’ project will begin by looking at Borovkov’s results and see if similar results can be shown in the case when R is irreducible. Chris will generalise these results to infinitely-many-colour urns, which leads to the natural and interesting open problem of making the number of colours go to infinity together with time.