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

Pólya骨灰盒是一个经典的离散时间随机过程，它描述了骨灰盒中包含不同颜色球的内容。在每个时间步，在瓮中均匀随机地选择一个球，并用一组新球替换到瓮中，这些新球的数量和颜色取决于所选球的颜色和替换规则，替换规则被编码为矩阵R。R是单位矩阵或不可约矩阵的情况在文献中得到了很好的研究，极限定理表明了当时间趋于无穷时瓮的组成是如何变化的。不可约的情况是经典的，可以追溯到马尔可夫在1906年的一些工作，从那以后得到了广泛的研究。不可简化的案例是最近的，Athreya和Karlin（1968）以及Janson（2004）发表了具有里程碑意义的论文。在单位矩阵的情况下，Borovkov最近证明了当初始球的数量随时间趋近于无穷大时瓮的组成的极限定理（见arXiv:1912.09665）。Borovkov的结果表明，不同行为之间存在过渡，这取决于两个因素（时间和瓮中球的初始数量）的缩放。本博士课题旨在证明替换矩阵不可约情况下的类似结果。由于在经典情况下，当瓮中球的初始数目固定时，不可约情况和恒等情况具有完全不同的行为，我们期望本博士的结果与Borovkov的结果有很大的不同。证明的方法也将与Borovkov的方法不同：我们相信他们将依赖于推广Athreya和Karlin（1968）以及最近的Janson（2004）在经典案例中使用的方法。作为实现这一目标的第一步，克里斯将从更简单的“平衡”情况开始，即在任何时候瓮中的总数都是确定的。这个案例在文献中很经典；我们希望它的分析将对更一般的非平衡情况提供见解。在解决了第一个问题之后，Chris将看看颜色的数量（不仅仅是最初的球的数量）随着时间的推移趋于无穷大的情况

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