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October 2020 Stochastic approximation on noncompact measure spaces and application to measure-valued Pólya processes
Cécile Mailler, Denis Villemonais
Ann. Appl. Probab. 30(5): 2393-2438 (October 2020). DOI: 10.1214/20-AAP1561

Abstract

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 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 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.

Citation

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Cécile Mailler. Denis Villemonais. "Stochastic approximation on noncompact measure spaces and application to measure-valued Pólya processes." Ann. Appl. Probab. 30 (5) 2393 - 2438, October 2020. https://doi.org/10.1214/20-AAP1561

Information

Received: 1 September 2018; Revised: 1 July 2019; Published: October 2020
First available in Project Euclid: 15 September 2020

MathSciNet: MR4149532
Digital Object Identifier: 10.1214/20-AAP1561

Subjects:
Primary: 60J25 , 60J80 , 62L20

Keywords: infinitely-many colors Pólya urns , Markov processes , Quasi-stationary distributions , stochastic approximation

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 5 • October 2020
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