Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas decroissant
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 \((Y_ n^{\theta})_{n\geq 0}\).
pathwise asymptotic behaviour of stochastic algorithms
scientific article; zbMATH DE number 3945179
scientific article