不定常波利亚瓮
Time-dependent Pólya urn
摘要：
我们考虑一个不定常版本的波利亚瓮包含黑色和白色的球。每次$n$从骨灰盒中随机抽取一个球，并与相同颜色的$σ_n$附加球一起放入骨灰盒中。白球的比例几乎必然收敛于随机极限$Θ$，而$\mathcal{D}=\{Θ\in\{0,1\}\}$表示当其中一种颜色占主导地位时的事件。用R. 1990年的佩曼特尔。我们描述的情况下${\mathbb P}(\mathcal{D})=0$和${\mathbb P}(\mathcal{D})>0$之间的相变。此外，我们研究了较强的垄断事件$\mathcal{M}$时，颜色之一最终停止再现，并分析制度${\mathbb P}(\mathcal{M})=0$，${\mathbb P}(\mathcal{M})\in (0,1)$，和${\mathbb P}(\mathcal{M})=1$之间的相变。
英文摘要：
We consider a time-dependent version of a Pólya urn containing black and white balls. At each time $n$ a ball is drawn from the urn at random and replaced in the urn along with $σ_n$ additional balls of the same colour. The proportion of white balls converges almost surely to a random limit $Θ$, and $\mathcal{D}=\{Θ\in\{0,1\}\}$ denotes the event when one of the colours dominates. The phase transition, in terms of the sequence $(σ_n)$, between the regimes ${\mathbb P}(\mathcal{D})=1$ and ${\mathbb P}(\mathcal{D})<1$ was obtained by R. Pemantle in 1990. We describe the phase transition between the cases ${\mathbb P}(\mathcal{D})=0$ and ${\mathbb P}(\mathcal{D})>0$. Further, we study the stronger monopoly event $\mathcal{M}$ when one of the colours eventually stops reappearing, and analyse the phase transition between the regimes ${\mathbb P}(\mathcal{M})=0$, ${\mathbb P}(\mathcal{M})\in (0,1)$, and ${\mathbb P}(\mathcal{M})=1$.
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