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

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A Pólya urn is a classical discrete-time stochastic process that describes the contents of an urn that contains balls ofdifferent colours. At each time step, a ball is chosen uniformly at random in the urn, and replaced into the urntogether with a set of new balls whose number and colours depend on the colour of the selected ball and on areplacement rule, which is encoded in a matrix R. The cases of R being either the identity matrix or irreducible arewell-studied in the literature and limiting theorems show how the composition of the urn behaves when time goes toinfinity. The irreducible case is classical and dates by to some work by Markov in 1906 and has been widely studiedsince 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 thenumber of initial balls goes to infinity together with time (see arXiv:1912.09665). Borovkov's results shows theexistence of a transition between different behaviours, depending on the scaling of the two factors (time and initialnumber of balls in the urn).This PhD project aims at proving analogous results for the case when the replacement matrix is irreducible. Becausethe irreducible and the identity case have drastically different behaviour in the classical case when the initial numberof balls in the urn is fixed, we expect the results of this PhD to be drastically different from Borovkov's. The methodsof proof will also be different from Borovkov's: we believe that they will rely on generalising the methods used in theclassical 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 inthe urn at all times is deterministic. This case is classical in the literature; we hope that its analysis will give insightinto 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 ofinitial balls) goes to infinity with time