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Une version fonctionnelle du théorème ergodique ponctuel. (Functional version of the pointwise ergodic theorem). (French) Zbl 0739.28007
Summary: Let \((x,\mathcal A ,T,\mu)\) be a measurable dynamical system and consider a non-empty subset \(S\) of \(L^ 1(\mu)\). We state sufficient conditions in order that the operator \(M(f)=\sup\left\{\sum^{n-1}_ 0 T^ kf/n,n\geqq 1\right\}\), indexed by \(S\), has a trajectorial version which is \(\mu\)-almost surely bounded or continuous on \(S\), when \(S\) is a non- empty subset of an Orlicz space \(L^{\varphi}(\mu)\). We also deduce a strengthening of the usual maximal inequalities regarding the averages \(\sum^{n-1}_ 0 T^ kf/n,n\geqq 1\), as well as a new result regarding strong integrability properties of stationary Gaussian processes.

MSC:
28D05 Measure-preserving transformations
60G15 Gaussian processes
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