Abstract
Let (Y,ℬ,μ,T) be an ergodic dynamical system. LetA be an nonempty subset ofL 2(μ) such that\(I(A) = \int_0^{diam(A)} {\sqrt {\log N(A,u)} du< \infty } \), whereA=sup{||sȒt||2μ ,s, t∈A} andN(A, u) is the smallest number ofL 2(μ)-open balls of radiusu, centered inA, enough to coverA. Let\(C(A) = \left\{ {\tfrac{1}{n}\Sigma _{i - 0}^{n - 1} f \circ T^i ,n \geqslant 1,f \in A} \right\}\). We prove as a consequence of a more general result, thatC(A) is aGB subset ofL 2(μ).
Article PDF
Similar content being viewed by others
References
Bourgain, J. (1988). Almost sure convergence and bounded entropy,Israel J. Math. 63, 79–95.
Fernique, X. (1974). Régularité des trajectoires de fonctions aléatoires gaussiennes,Lect. Note 480, 1–97.
Fernique, X. (1985). Gaussian random vectors and their reproducing kernel Hilbert space, Technical report No. 34, University of Ottawa.
Stein, E. M. (1961). On limits of sequences of operators,Ann. Math. 74, 140–170.
Talagrand, M. (1987). Regularity of Gaussian processes.Acta. Math. 159, 99–149.
Weber, M. (1990). Une version fonctionnelle du Théorème ergodique ponctuel,Comptes Rendus Acad. Sci. Paris. Sér. I 311, 131–133.
Weber, M. (1992). Méthodes de sommation matricielles,Comptes Rendus Acad. Sci. Paris, Sér. I 315, 759–764.
Weber, M. (1993). GC sets, Stein's elements and matrix summation methods, Prépublication IRMA No. 027.
Weber, M. (1994). GB and GC sets in ergodic theory, IXth Conference on Probability in Banach Spaces, Sandberg 1993, V. 35, Basel, Birkhäuser.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Weber, M. Coupling of the GB set property for ergodic averages. J Theor Probab 9, 105–112 (1996). https://doi.org/10.1007/BF02213736
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02213736