On the Rate of Convergence in the Central Limit Theorem for Martingales with Discrete and Continuous Time

Haeusler, Erich

doi:10.1214/aop/1176991901

Cited by 79 publications

(62 citation statements)

References 21 publications

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“…D'abord l'action de la transformation sur les fonctions régulières ne fait pas apparaître directement une suite de différences de martingale : il faut compter avec un terme pertubateur de type cobord. Ensuite les résultats de [2] et [13] comportent des conditions de contrôle non-stationnaires qui sontévidemment violées dans les cas qui nous intéressent. Par exemple, la vitesse en n − 1 2 prouvée dans [2] l'est sous une hypothèse portant sur les cubes des variables qui n'est généralement pas satisfaite dans notre cadre.…”

Section: Introductionunclassified

Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques

Borgne¹,

Pène²

2005

Bul. Soc. Math. France

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Résumé. -Nous présentons une méthode permettant d'établir le théorème limite central avec vitesse en n −1/2 pour certains systèmes dynamiques. Elle est basée sur une propriété de décorrélation forte qui semble assez naturelle dans le cadre des systèmes quasi-hyperboliques. Nous prouvons que cette propriété est satisfaite par les exemples des flots diagonaux sur un quotient compact de SL(d, R) et les « transformations » non uniformément hyperboliques du tore T 3é tudiées par Shub et Wilkinson.Abstract (Rate of convergence in the central limit theorem). -We present a method which enables to establish the central limit theorem with rate of convergence in n −1/2 for certain dynamical systems. It is based on a strong decorrelation property that seems to be quite natural for quasi-hyperbolic systems. We prove that this property is satisfied by the diagonal flows on a compact quotient of SL(d, R) and the non uniformly hyperbolic transformations of the torus T 3 studied by Shub and Wilkinson.

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Section: Introductionunclassified

Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques

Borgne¹,

Pène²

2005

Bul. Soc. Math. France

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“…We are now considering a stationary, ergodic diusion process de®ned by (11) with the stationary distribution m given by…”

Section: Asymptotic Expansion Of a Functional Of An Ergodic Diusionmentioning

confidence: 99%

“…This norm is independent of the choice of the bases fh 1Yi g and fh 2Yj g. r 1 r 2 denotes the set of bilinear forms v satisfying that jvj r 1 r 2`I , and is a Hilbert space equipped with the Hilbertian norm jvj r 1 r 2 . It is clear that Lemma 7 Let t X t P R satisfy the stochastic dierential equation (11). Suppose that the following conditons are satis®ed:…”

Section: Asymptotic Expansion Of a Functional Of An Ergodic Diusionmentioning

confidence: 99%

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Malliavin calculus and asymptotic expansion for martingales

Yoshida

1997

Probab Theory Relat Fields

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We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the CrameÂ r condition did in the case of independent observations. Applications to statistics are presented.Mathematics Subject Classi®cation (1991): 60F05, 60G44, 62E20

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“…In this paper we are interested in the problem of how to extend results on the rate of convergence in the martingale CLT proved under the condition (1.1) to the general case. In the one dimensional Hilbert space the problem has been solved by a simple stopping time technique (see, e.g., Bolthausen [1], Haeusler [4]). We present a method allowing one to attack the problem when the martingale takes values in either finite or infinite dimensional Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

On the conditional covariance condition in the martingale CLT

Račkauskas¹

1995

Lith Math J

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Abstract--The paper develops a method allowing one to figure out how a convergence rate in the martingale central limit theorem depends on the conditional covariance structure of the martingale. The method is based on constructing "stopping projections" that control the behavior of the conditional covariances of martingale differences. A discrete time martingale taking values in either finite or infinite dimensional Hilbert space is considered.

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On the Rate of Convergence in the Central Limit Theorem for Martingales with Discrete and Continuous Time

Cited by 79 publications

References 21 publications

Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques

Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques

Malliavin calculus and asymptotic expansion for martingales

On the conditional covariance condition in the martingale CLT

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