The Annals of Applied Probability

On the stability of some controlled Markov chains and its applications to stochastic approximation with Markovian dynamic

Christophe Andrieu, Vladislav B. Tadić, and Matti Vihola

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Abstract

We develop a practical approach to establish the stability, that is, the recurrence in a given set, of a large class of controlled Markov chains. These processes arise in various areas of applied science and encompass important numerical methods. We show in particular how individual Lyapunov functions and associated drift conditions for the parametrized family of Markov transition probabilities and the parameter update can be combined to form Lyapunov functions for the joint process, leading to the proof of the desired stability property. Of particular interest is the fact that the approach applies even in situations where the two components of the process present a time-scale separation, which is a crucial feature of practical situations. We then move on to show how such a recurrence property can be used in the context of stochastic approximation in order to prove the convergence of the parameter sequence, including in the situation where the so-called stepsize is adaptively tuned. We finally show that the results apply to various algorithms of interest in computational statistics and cognate areas.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 1 (2015), 1-45.

Dates
First available in Project Euclid: 16 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1418740177

Digital Object Identifier
doi:10.1214/13-AAP953

Mathematical Reviews number (MathSciNet)
MR3297764

Zentralblatt MATH identifier
1317.65004

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 60J05: Discrete-time Markov processes on general state spaces

Keywords
Stability Markov chains stochastic approximation controlled Markov chains adaptive Markov chain Monte Carlo

Citation

Andrieu, Christophe; Tadić, Vladislav B.; Vihola, Matti. On the stability of some controlled Markov chains and its applications to stochastic approximation with Markovian dynamic. Ann. Appl. Probab. 25 (2015), no. 1, 1--45. doi:10.1214/13-AAP953. https://projecteuclid.org/euclid.aoap/1418740177


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