非紧测度空间上的随机逼近及其对测度值Pólya过程的应用
Stochastic approximation on non-compact measure spaces and application to measure-valued Pólya processes
摘要：
本文的主要结果是证明了定义在非紧空间上的测度空间上的随机逼近算法的几乎必然收敛。我们的动机是将这一结果应用于测度值Pólya过程（MVPPs，也称为无限多Pólya瓮）。我们的主要思想是使用福斯特-李雅普诺夫型准则在一个新的方式推广随机逼近方法的测度值马尔可夫过程与非紧的底层空间，克服在一个相当普遍的情况下，现有的研究这个问题的主要困难之一。从MVPPs的角度来看，我们的结果意味着几乎必然收敛的一个大类的MVPPs，这种收敛到现在为止，只有获得特定的例子，只有依概率收敛建立一般类。此外，我们的方法允许我们通过向无限多色骨灰盒的不同颜色添加“权重”来扩展MVPPs的定义。我们还展示了非“平衡”MVPPs和准平稳分布的马尔可夫过程，这使我们能够治疗之间的联系，在文献中的第一次，非平衡的情况下。最后，我们展示了我们的结果可以应用于设计随机近似算法的近似的准平稳分布的离散和连续时间的马尔可夫过程的非紧空间。
英文摘要：
Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a non-compact space. Our motivation is to apply this result to measure-valued Pólya processes (MVPPs, also known as infinitely-many Pólya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochastic-approximation methods to measure-valued Markov processes with a non-compact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs, this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding "weights" to the different colors of the infinitely-many-color urn. We also exhibit a link between non-"balanced" MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the non-balanced case. Finally, we show how our result can be applied to designing stochastic-approximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on non-compact spaces.
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。
