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Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas décroissant
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  • Published: September 1987

Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas décroissant

  • M. Métivier1 &
  • P. Priouret2 

Probability Theory and Related Fields volume 74, pages 403–428 (1987)Cite this article

  • 153 Accesses

  • 37 Citations

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Summary

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 hypotheses allowing discontinuities on the adaptation termf. The process (Y n) n≧0 is a Markov chain “controlled by (θ n )”. For each θ fixed the Markov chain (Y θn ) n≧0 is essentially of positive recurrent type.

A first theorem generalizes to this situation a Ljung's theorem (L. Ljung, IEEE Trans. AC 22, 2, p. 551–575 (1977)).

An almost sure convergence theorem is proved under the existence of a global Lyapunov function for the associated deterministic differential equation.

$$\frac{{d\bar \theta (t)}}{{d(t)}} = h(\bar \theta (t))$$

whereh(θ) = ∫f(θ, y) Γ θ (d, y) and Γ θ is the invariant probability of the Markov chain (Y θn ) n≧0.

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Author information

Authors and Affiliations

  1. Centre de Mathématiques Appliquées, École Polytechnique, F-91128, Palaiseau, France

    M. Métivier

  2. Laboratoire de Probabilités, Université Pierre et Marie Curie, F-75230, Paris, France

    P. Priouret

Authors
  1. M. Métivier
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  2. P. Priouret
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Métivier, M., Priouret, P. Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas décroissant. Probab. Th. Rel. Fields 74, 403–428 (1987). https://doi.org/10.1007/BF00699098

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  • Received: 22 December 1984

  • Revised: 18 June 1986

  • Issue Date: September 1987

  • DOI: https://doi.org/10.1007/BF00699098

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