论文标题
通过随机近似阳性加强的广义时间依赖性Pólyaurns
Positive reinforced generalized time-dependent Pólya urns via stochastic approximation
论文作者
论文摘要
考虑一个广义的时间依赖性的Pólyaurn过程，定义为如下。令$ d \ in \ mathbb {n} $为urns/颜色的数量。在每次$ n $时，我们将$σ_n$球随机分发到$ d $ urns，成比例地分配给$ f $，其中$ f $是有效的增强功能。我们考虑一般的积极强化功能$ \ MATHCAL {r} $假设有一定的单调性和生长条件。类$ \ Mathcal {r} $包括凸函数和经典的情况$ f（x）= x^α$，$α> 1 $。该论文的新颖性在于将随机近似技术扩展到$ d $维情况，并证明该过程最终将固定在某些随机的urn，而其他urn则将不再接收任何球。
Consider a generalized time-dependent Pólya urn process defined as follows. Let $d\in \mathbb{N}$ be the number of urns/colors. At each time $n$, we distribute $σ_n$ balls randomly to the $d$ urns, proportionally to $f$, where $f$ is a valid reinforcement function. We consider a general class of positive reinforcement functions $\mathcal{R}$ assuming some monotonicity and growth condition. The class $\mathcal{R}$ includes convex functions and the classical case $f(x)=x^α$, $α>1$. The novelty of the paper lies in extending stochastic approximation techniques to the $d$-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls any more.
