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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