On the distribution function of the majorant of ergodic means

Epremidze, Lasha

Studia Mathematica, Tome 103 (1992), p. 1-15 / Harvested from The Polish Digital Mathematics Library

Résumé

Let T be a measure-preserving ergodic transformation of a measure space (X,,μ) and, for f ∈ L(X), let $f * = s u p_{N} 1 / N \sum_{m = 0}^{N - 1} f \circ T^{m}$ . In this paper we mainly investigate the question of whether (i) $ʃ_{a}^{\infty} | μ (f * > t) - 1 / t ʃ_{(f * > t)} f d μ | d t < \infty$ and whether (ii) $ʃ_{a}^{\infty} | μ (f * > t) - 1 / t ʃ_{(f > t)} f d μ | d t < \infty$ for some a > 0. It is proved that (i) holds for every f ≥ 0. (ii) holds if f ≥ 0 and f log log (f + 3) ∈ L(X) or if μ(X) = 1 and the random variables $f \circ T^{m}$ are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries.

Publié le : 1992-01-01

Zbl 0809.28011

EUDML-ID : urn:eudml:doc:215932

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     author = {Lasha Epremidze},
     title = {On the distribution function of the majorant of ergodic means},
     journal = {Studia Mathematica},
     volume = {103},
     year = {1992},
     pages = {1-15},
     zbl = {0809.28011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv103i1p1bwm}
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Epremidze, Lasha. On the distribution function of the majorant of ergodic means. Studia Mathematica, Tome 103 (1992) pp. 1-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv103i1p1bwm/

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