Some optimal pointwise ergodic theorems with rate

Christophe Cuny

doi:10.1016/j.crma.2009.04.034

Dynamical Systems/Probability Theory

Some optimal pointwise ergodic theorems with rate
[Théorèmes ergodiques ponctuels avec vitesse optimale]

Présenté par : Wendelin Werner

Christophe Cuny ¹

¹ Université de la Nouvelle-Calédonie, Équipe ERIM, B.P. R4, 98800 Nouméa, New Caledonia

Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 953-958.

Résumé
Abstract

Soit T un opérateur de Dunford–Schwartz sur l'espace de probabilité $(X, Σ, μ)$ et $p > 1$ . Pour f dans l'image d'opérateurs judicieux de $L^{p} (X, Σ, μ)$ , nous obtenons des théorèmes ergodiques ponctuels avec vitesse, par une méthode due à Derriennic et Lin (2001). Lorsque T est induit par une transformation préservant μ, nous montrons l'optimalité des résultats, la preuve étant inspirée par une construction de Déniel (1989).

Let T be a Dunford–Schwartz operator on the probability space $(X, Σ, μ)$ and $p > 1$ . For f in the range of suitable operators of $L^{p} (X, Σ, μ)$ , we obtain pointwise ergodic theorems with rate, using a method of Derriennic and Lin (2001). When T is induced by a μ-preserving transformation, these results are shown to be optimal. The proof of the optimality is inspired from a construction of Déniel (1989).

Reçu le : 2009-02-15
Accepté le : 2009-04-29
Publié le : 2009-06-03

DOI : 10.1016/j.crma.2009.04.034

Affiliations des auteurs :

Christophe Cuny ¹

¹ Université de la Nouvelle-Calédonie, Équipe ERIM, B.P. R4, 98800 Nouméa, New Caledonia

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Christophe Cuny. Some optimal pointwise ergodic theorems with rate. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 953-958. doi : 10.1016/j.crma.2009.04.034. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.034/

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A. G. Kachurovskii; I. V. Podvigin Measuring the rate of convergence in the Birkhoff ergodic theorem, Mathematical Notes, Volume 106 (2019) no. 1, pp. 52-62 | DOI:10.1134/s0001434619070058 | Zbl:1427.37004
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Christophe Cuny; Michael Lin Limit theorems for Markov chains by the symmetrization method, Journal of Mathematical Analysis and Applications, Volume 434 (2016) no. 1, pp. 52-83 | DOI:10.1016/j.jmaa.2015.07.061 | Zbl:1334.60035
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Christophe Cuny; Magda Peligrad Central limit theorem started at a point for stationary processes and additive functionals of reversible Markov chains, Journal of Theoretical Probability, Volume 25 (2012) no. 1, pp. 171-188 | DOI:10.1007/s10959-010-0321-8 | Zbl:1247.60031
Florence Merlevède; Costel Peligrad; Magda Peligrad Almost sure invariance principles via martingale approximation, Stochastic Processes and their Applications, Volume 122 (2012) no. 1, pp. 170-190 | DOI:10.1016/j.spa.2011.09.004 | Zbl:1230.60029
Christophe Cuny Pointwise ergodic theorems with rate with applications to limit theorems for stationary processes, Stochastics and Dynamics, Volume 11 (2011) no. 1, pp. 135-155 | DOI:10.1142/s0219493711003206 | Zbl:1210.60031
Dalibor Volný Martingale approximation and optimality of some conditions for the central limit theorem, Journal of Theoretical Probability, Volume 23 (2010) no. 3, pp. 888-903 | DOI:10.1007/s10959-010-0275-x | Zbl:1205.60075

Cité par 8 documents. Sources : Crossref, zbMATH

^☆ Research partially carried out at Ben-Gurion University, supported by its Center for Advanced Studies in Mathematics.

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