论文标题
pólyaurns的功能极限定理，具有生长的初始组成
Functional limit theorems for Pólya urns with growing initial compositions
论文作者
论文摘要
在本文中，我们证明了pólyaurn过程的功能极限定理，其绘制数量和初始球数倾向于无穷大。这是由Borovkov [5]的最新工作引起的，当urn具有身份替代规则时，它们证明了该模型的功能极限定理。我们将此结果推广到任意平衡的替换规则（添加到urn的球的总数是确定性的）。可能会根据如何使初始球的数量与urn的抽签数量允许初始球的数量缩放。在每个制度中，我们都会显示一个一阶确定性限制和高斯二阶波动，其中这些限制过程的行为取决于制度，urn的初始组成和urns替换规则。 为了证明我们的主要结果，我们将过程嵌入了连续时间并使用Martingale理论。尽管这些方法是古典的，因为Athreya＆Karlin [1]和Janson [11]的作品，我们最初增长的构图的设置需要许多新的想法。在我们设置中连续嵌入的限制结果的主要区别在于，当初始组合物与抽奖次数相比，分支过程没有时间达到平衡。因此，将结果转回离散时间也比Janson [11]要困难得多。有趣的是，我们的连续时间结果在替换结构的较弱的假设下与多类分支过程的经典结果相比。特别是，我们不需要任何“不可约性”假设。
In this paper, we prove functional limit theorems for Pólya urn processes whose number of draws and initial number of balls tend to infinity together. This is motivated by recent work of Borovkov [5], where they prove a functional limit theorem for this model when the urn has identity replacement rule. We generalize this result to arbitrary balanced replacement rules (the total number of balls added to the urn is deterministic). Three asymptotic regimes are possible depending on how one lets the number of initial balls scale with the number of draws of the urn. In each regime, we show a first order deterministic limit and Gaussian second order fluctuations, where the behaviour of these limit processes depend on the regime, the initial composition of the urn, and the urns replacement rule. To prove our main results, we embed the process in continuous-time and use martingale theory. Although these methods are classical since the works of Athreya & Karlin [1] and Janson [11], our setting with initial growing composition necessitates many new ideas. The main difference in proving limiting results for the continuous time embedding in our setting is that, when the initial composition is large compared to the number of draws, the branching process does not have time to reach equilibrium. Because of this, translating the results back to discrete-time is also much harder than in Janson [11]. Interestingly, our continuous-time results hold under weaker assumptions on the replacement structure than classical results for multi-type branching processes; in particular, we do not need any "irreducibility" assumption.
