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Coupling of the GB set property for ergodic averages. (English) Zbl 0851.60037
This article pursues a connection first made by J. Bourgain [Isr. J. Math. 63, No. 1, 79-97 (1988; Zbl 0677.60042)] between dynamical systems and the notion of GB-sets within a Hilbert space, a notion of compactness motivated by the theory of Gaussian processes. More specifically, for a Hilbert space \(H\), there exists an isometry to a linear space (with covariance as inner-product) of mean-0 Gaussian random variables \((Z_a, a \in H)\), the isonormal process defined on a canonical probability space. A subset \(A \subset H\) is called a GB-set if \(E (\sup_{a \in A} Z_a) < \infty\). Bourgain’s result had said that if \(\{S_n, n \geq 1\}\) is a family of bounded operators on \(L^2 (\Omega, {\mathcal B}, \mu)\) which commute with \(T\), where \((\Omega, {\mathcal B}, \mu,T)\) is an ergodic dynamical system, and if \(p \geq 2\) is such that \(\mu (\{x \in \Omega : \sup_n |S_n (f)(x) |< \infty\}) = 1\) for all \(f \in L^p (\mu)\), then for each such \(f\), \(\{S_n (f), n \geq 1\}\) is a GB-set. This article uses techniques of Bourgain and E. Stein to extend this result, to say that if the operators \(S_n\) additionally satisfy an \(L^p\) maximal inequality for some \(p \geq 2\), and if \(A \subset L^2 (\mu)\) is nonempty with a finite metric-entropy integral, then \(\{S_n(f), f \in A, n \geq 1\}\) is again a GB-set. The main application of the result is to ergodic-average operators \(S_n\) defined by \(S_n (f) = n^{-1} \sum^n_{i = 1} f \circ T^i\).

MSC:
60G15 Gaussian processes
60F15 Strong limit theorems
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[1] Bourgain, J. (1988). Almost sure convergence and bounded entropy,Israel J. Math. 63, 79–95. · Zbl 0677.60042
[2] Fernique, X. (1974). Régularité des trajectoires de fonctions aléatoires gaussiennes,Lect. Note 480, 1–97.
[3] Fernique, X. (1985). Gaussian random vectors and their reproducing kernel Hilbert space, Technical report No. 34, University of Ottawa. · Zbl 0596.60009
[4] Stein, E. M. (1961). On limits of sequences of operators,Ann. Math. 74, 140–170. · Zbl 0103.08903
[5] Talagrand, M. (1987). Regularity of Gaussian processes.Acta. Math. 159, 99–149. · Zbl 0712.60044
[6] Weber, M. (1990). Une version fonctionnelle du Théorème ergodique ponctuel,Comptes Rendus Acad. Sci. Paris. Sér. I 311, 131–133. · Zbl 0739.28007
[7] Weber, M. (1992). Méthodes de sommation matricielles,Comptes Rendus Acad. Sci. Paris, Sér. I 315, 759–764. · Zbl 0768.40002
[8] Weber, M. (1993). GC sets, Stein’s elements and matrix summation methods, Prépublication IRMA No. 027.
[9] Weber, M. (1994). GB and GC sets in ergodic theory, IXth Conference on Probability in Banach Spaces, Sandberg 1993, V. 35, Basel, Birkhäuser. · Zbl 0808.28011
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