# On the distribution function of the majorant of ergodic means

Studia Mathematica (1992)

- Volume: 103, Issue: 1, page 1-15
- ISSN: 0039-3223

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topEpremidze, Lasha. "On the distribution function of the majorant of ergodic means." Studia Mathematica 103.1 (1992): 1-15. <http://eudml.org/doc/215932>.

@article{Epremidze1992,

abstract = {Let T be a measure-preserving ergodic transformation of a measure space (X,,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.},

author = {Epremidze, Lasha},

journal = {Studia Mathematica},

keywords = {distribution function; majorant of ergodic means; measure-preserving ergodic transformation; inequalities},

language = {eng},

number = {1},

pages = {1-15},

title = {On the distribution function of the majorant of ergodic means},

url = {http://eudml.org/doc/215932},

volume = {103},

year = {1992},

}

TY - JOUR

AU - Epremidze, Lasha

TI - On the distribution function of the majorant of ergodic means

JO - Studia Mathematica

PY - 1992

VL - 103

IS - 1

SP - 1

EP - 15

AB - Let T be a measure-preserving ergodic transformation of a measure space (X,,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.

LA - eng

KW - distribution function; majorant of ergodic means; measure-preserving ergodic transformation; inequalities

UR - http://eudml.org/doc/215932

ER -

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