On positively expansive differentiable maps

Kazuhiro Sakai

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Mild law of large numbers and its consequences

Jaroslav Mohapl (1993)

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On pointwise ergodic theorems for positive operators

Ryotaro Sato (1990)

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On the distribution function of the majorant of ergodic means

Lasha Epremidze (1992)

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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. ...