On the almost sure asymptotic behaviour of stochastic algorithms

Pelletier, Mariane

doi:10.1016/s0304-4149(98)00029-5

Cited by 51 publications

(63 citation statements)

References 15 publications

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“…Such a weighted empirical mean also appears in [16], where we proved, under assumptions (A1) to (A3), the following quadratic strong law of large numbers:…”

Section: Assumptions and Main Resultsmentioning

confidence: 58%

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An Almost Sure Central Limit Theorem for Stochastic Approximation Algorithms

Pelletier¹

1999

Journal of Multivariate Analysis

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We prove an almost sure central limit theorem for some multidimensional stochastic algorithms used for the search of zeros of a function and known to satisfy a central limit theorem. The almost sure version of the central limit theorem requires either a logarithmic empirical mean (in the same way as in the case of independent identically distributed variables) or another scale, depending on the choice of the algorithm gains. Academic PressAMS 1991 subject classifications: 62L20, 62F12, 60F05, 60F15.

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“…Such a weighted empirical mean also appears in [16], where we proved, under assumptions (A1) to (A3), the following quadratic strong law of large numbers:…”

Section: Assumptions and Main Resultsmentioning

confidence: 58%

“…By following a method often used by Lai and Wei (see for instance [12]), we proved in [16] that, in order to establish an almost sure asymptotic property of the algorithm (1) on 1(z*), we can strengthen assumptions (A2), assuming that almost surely on the whole set 0,…”

Section: Assumptions and Main Resultsmentioning

confidence: 99%

An Almost Sure Central Limit Theorem for Stochastic Approximation Algorithms

Pelletier¹

1999

Journal of Multivariate Analysis

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“…Pour des travaux antérieurs sur la loi forte quadratique des grands nombres LFQ relatifs aux martingales unidimensionnellesà temps discret ou continu, on pourra consulter [3,4,9,19,21,24]... Dans [22], une loi LFQ aétéétablie pour des algorithmes stochastiques vectoriels.…”

Section: Commentaires Bibliographiquesunclassified

“…[12,22]) ; donc ii) est vérifiée. On en déduit i) et iii) car sous ces hypothèses la propriété (5.10) a lieu.…”

Section: ) Preuves Des Lfq Sous Les Conditions (C) Ou (C )unclassified

Théorèmes limites avec poids pour les martingales vectorielles

Chaabane

Maaouia

2000

ESAIM: PS

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Abstract.We give limit theorems specifying weak and strong rates of convergence associated to a quadratic extension of the martingale almost-sure central limit theorem. Some typical examples are discussed to illustrate how to make use of them in statistic.Résumé. On donne des théorèmes limites précisant les vitesses de convergence (en loi et au sens presque-sûr) associéesà une extension quadratique du théorème de limite centrale presque-sûre pour les martingales discrètes vectorielles. Des exemples typiques illustrent l'usage qu'on peut en faire en statistique.

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“…The asymptotic normality of θ n may be found in [17] whereas the quadratic strong law and the law of iterated logarithm are established in [16]. Results for randomly truncated version of the Robbins-Monro algorithm are given in [13].…”

Section: Introductionmentioning

confidence: 99%

Recursive estimation in a class of models of deformation

Fraysse

2014

Journal of Statistical Planning and Inference

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Abstract. The paper deals with the statistical analysis of several data sets associated with shape invariant models with different translation, height and scaling parameters. We propose to estimate these parameters together with the common shape function. Our approach extends the recent work of Bercu and Fraysse to multivariate shape invariant models. We propose a very efficient Robbins-Monro procedure for the estimation of the translation parameters and we use these estimates in order to evaluate scale parameters. The main pattern is estimated by a weighted Nadaraya-Watson recursive estimator. We provide almost sure convergence and asymptotic normality for all estimators. Finally, we illustrate the convergence of our estimation procedure on simulated data as well as on real ECG data.

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On the almost sure asymptotic behaviour of stochastic algorithms

Cited by 51 publications

References 15 publications

An Almost Sure Central Limit Theorem for Stochastic Approximation Algorithms

An Almost Sure Central Limit Theorem for Stochastic Approximation Algorithms

Théorèmes limites avec poids pour les martingales vectorielles

Recursive estimation in a class of models of deformation

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