Limit theorems for Pólya urns with initial composition tending to infinity with time

A Pólya urn is a classical discrete-time stochastic process that describes 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. The irreducible case is classical and dates by to some work by Markov in 1906 and has been widely studied
since then. The irreducible case is more recent, with landmark papers by Athreya and Karlin (1968), and Janson
(2004).
In the case of the identity matrix, Borovkov recently proved limiting theorems for the composition of the urn when the
number of initial balls goes to infinity together with time (see arXiv:1912.09665). Borovkov's results shows the
existence of a transition between different behaviours, depending on the scaling of the two factors (time and initial
number of balls in the urn).
This PhD project aims at proving analogous results for the case when the replacement matrix is irreducible. Because
the irreducible and the identity case have drastically different behaviour in the classical case when the initial number
of balls in the urn is fixed, we expect the results of this PhD to be drastically different from Borovkov's. The methods
of proof will also be different from Borovkov's: we believe that they will rely on generalising the methods used in the
classical case by Athreya and Karlin (1968), and more recently Janson (2004).
As a first step towards this goal, Chris will start by looking at the simpler ``balanced'' case when the total number in
the urn at all times is deterministic. This case is classical in the literature; we hope that its analysis will give insight
into t he more general non-balanced case.
After solving this first question, Chris will look at the case when the number of colours (and not only the number of
initial balls) goes to infinity with time

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

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- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

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