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On the distribution function of the majorant of ergodic means

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Epremidze L.

Studia Mathematica

1992

tom 103

nr 1

1-15

Let T be a measure-preserving ergodic transformation of a measure space (X,𝕊,μ) and, for f ∈ L(X), let $f* = sup_N {1/N} ∑_{m=0}^{N - 1} f ∘ T^m$. In this paper we mainly investigate the question of whether (i) $ʃ_a^∞ |μ(f*>t) - {1/t} ʃ_{(f*>t)} fdμ|dt < ∞$ and whether (ii) $ʃ_a^∞ |μ(f*>t) - {1/t} ʃ_{(f>t)} fdμ|dt < ∞$ 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.

Ograniczanie wyników

1 Studia Mathematica

1 Epremidze L.

1 1992

Wyniki wyszukiwania

On the distribution function of the majorant of ergodic means