循环瓮合成的加细渐近性
Refined Asymptotics for the Composition of Cyclic Urns
摘要：
循环骨灰盒是用于$0,\ldots,m-1$类型的球的骨灰盒模型。骨灰盒在时间零处以初始配置开始。然后，在每个时间步中，首先从瓮中均匀地且独立于过去地抽取一个球。如果它的类型是$j$，则它与类型为$j+1 \mod m$的新球一起返回到骨灰盒。案例$m=2$是著名的弗里德曼骨灰盒。合成向量，即，在$n$步骤之后的每种类型的球的数量的矢量在归一化之后已知对于$2\le m\le 6$是渐近正态的。对于$m\ge 7$，已知归一化的组成向量不收敛。然而，有一个几乎肯定的近似周期随机向量。在本文中，这个周期随机向量的渐近波动被确定。我们表明，这些波动是渐近正常的所有$7\le m\le 12$。对于$m\ge 13$，我们也发现渐近正常的波动时，在一个更精细的方式规范化。只有当$6$不分裂$m$时，这些波动才具有最大尺寸$m-1$。对于$m$是$6$的倍数，波动由二维子空间支持。
英文摘要：
A cyclic urn is an urn model for balls of types $0,\ldots,m-1$. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is $j$, it is then returned to the urn together with a new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after $n$ steps is, after normalization, known to be asymptotically normal for $2\le m\le 6$. For $m\ge 7$ the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector. In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all $7\le m\le 12$. For $m\ge 13$ we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations are of maximal dimension $m-1$ only when $6$ does not divide $m$. For $m$ being a multiple of $6$ the fluctuations are supported by a two-dimensional subspace.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
