Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas decroissant
type: Item
instance of: scholarly article
title: Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas decroissant
zbMATH Open document ID: 0588.62153
DOI: 10.1007/BF00699098
publication date: 1987-01-01T00:00:00Z
review text: The paper studies the pathwise asymptotic behaviour of stochastic algorithms of the following general form \[ \theta_{n+1}=\theta_ n+\gamma_{n+1}f(\theta_ n,Y_{n+1}), \] the hypothesis allowing discontinuities on the adaptation term f. The process \((Y_ n)_{n\geq 0}\) is a Markov chain ''controlled by \((\theta_ n)''\). For each \(\theta\) fixed the Markov chain \((Y_ n^{\theta})_{n\geq 0}\) is essentially of positive recurrent type. A first theorem generalizes to this situation a theorem of \textit{L. Ljung} [IEEE Trans. Autom. Control AC-22, 551-575 (1977; Zbl 0362.93031)]. An almost sure convergence theorem is proved under the existence of a global Lyapunov function for the associated deterministic differential equation d\({\bar \theta}\)(t)/dt\(=h({\bar \theta}(t))\) where \(h(\theta)=\int f(\theta,y)\Gamma_{\theta}(dy)\) and \(\Gamma_{\theta}\) is the invariant probability of the Markov chain \((Y_ n^{\theta})_{n\geq 0}\).
Mathematics Subject Classification ID: 62L20
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 62L12
zbMATH DE Number: 3945179
zbMATH Keywords: pathwise asymptotic behaviour of stochastic algorithms
zbMATH Keywords: Markov chain
zbMATH Keywords: almost sure convergence theorem
zbMATH Keywords: global Lyapunov function
zbMATH Keywords: positive recurrence
author: Michel Métivier
author: P. Priouret
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00699098
OpenAlex ID: W2090340215
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3945179