基于随机逼近的正强化广义不定常Pólya壶
Positive reinforced generalized time-dependent Pólya urns via stochastic approximation
摘要：
考虑一个广义的不定常波利亚翁过程，定义如下。设$d\in \mathbb{N}$为瓮/颜色的数量。在每个时间$n$，我们将$σ_n$球随机分配到$d$瓮，与$f$成比例，其中$f$是有效的强化函数。我们考虑了一类一般的正强化函数$\mathcal{R}$，并假设其具有一定的单调性和增长性。类$\mathcal{R}$包括凸函数和经典情形$f(x)=x^α$、$α>1$。本文的新颖之处在于扩展随机逼近技术的$d$维的情况下，并证明，最终的过程将固定在一些随机瓮和其他瓮将不会收到任何球了。
英文摘要：
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.
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。
