STOCHASTIC APPROXIMATION ON NONCOMPACT MEASURE SPACES AND APPLICATION TO MEASURE-VALUED POLYA PROCESSES

• 成果类型: Article
• 署名作者: Mailler, Cecile; Villemonais, Denis
• 署名单位: University of Bath; Inria; Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1561
• 发表日期: 2020
• 页码: 2393-2438
• 关键词: quasi-stationary distributions; LIMIT-THEOREMS; branching-processes; CONVERGENCE; simulation; algorithm; DYNAMICS; points; models; nodes

摘要

Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a noncompact space. Our motivation is to apply this result to measure-valued Polya processes (MVPPs, also known as infinitely-many Polya 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 noncompact 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 nonbalanced 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 noncompact spaces.