Displaying similar documents to “On metric diophantine approximation and subsequence ergodic theory.”

On the distribution function of the majorant of ergodic means

Lasha Epremidze (1992)

Studia Mathematica

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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 ∑ m = 0 N - 1 f ∘ T m . In this paper we mainly investigate the question of whether (i) ʃ a ∞ | μ ( f * > t ) - 1 / t ʃ ( f * > t ) f d μ | d t < ∞ and whether (ii) ʃ a ∞ | μ ( f * > t ) - 1 / t ʃ ( f > t ) f d μ | d t < ∞ 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 ∘ T m are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries. ...