论文标题
随机URN模型与不可还原且可还原的替换政策的收敛性
Convergence of randomized urn models with irreducible and reducible replacement policy
论文作者
论文摘要
概括性理论中考虑的最简单，最有用的模型之一。由于Athreya和Ney（1972）表明，在$ l \ log l $时刻假设下的随机urn模型中，在随机的urn模型中几乎确定了urn比例的收敛性，因此该假设被认为是最弱的时刻假设，但从未显示过必要的假设。在本文中，我们研究了广义弗里德曼urn的强大和弱收敛性。事实证明，当随机替换矩阵的概率上是不可约的时，对于几乎确定的urn比例收敛的足够和必要的时刻假设是，对替换矩阵的期望是有限的，它比$ l \ log l $ somks pastion and replocement and replocement and repotige nofe figest是$ l \ l \ l \ f \ l \ r \ f \ f \ f \ l \ f \ l \ f \ f \ l \ f \ \ f \ \时刻。也得出了非殖民概括的弗里德曼骨灰酶的收敛速度以及强度和弱收敛性。
Generalized Friedman urn is one of the simplest and most useful models considered in probability theory. Since Athreya and Ney (1972) showed the almost sure convergence of urn proportions in a randomized urn model with irreducible replacement matrix under the $L\log L$ moment assumption, this assumption has been regarded as the weakest moment assumption, but the necessary has never been shown. In this paper, we study the strong and weak convergence of generalized Friedman urns. It is proved that, when the random replacement matrix is irreducible in probability, the sufficient and necessary moment assumption for the almost sure convergence of the urn proportions is that the expectation of the replacement matrix is finite, which is less stringent than the $L\log L$ moment assumption, and when the replacement is reducible, the $L\log L$ moment assumption is the weakest sufficient condition. The rate of convergence and the strong and weak convergence of non-homogenous generalized Friedman urns are also derived.
