On the distribution function of the majorant of ergodic means

Lasha Epremidze

Studia Mathematica (1992)

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

Abstract

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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.

How to cite

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Epremidze, 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 -

References

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  13. [13] R. Sato, On the ratio maximal function for an ergodic flow, Studia Math. 80 (1984), 129-139. Zbl0521.28015
  14. [14] O. Tsereteli, On the distribution function of the conjugate function of a nonnegative Borel measure, Trudy Tbilis. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 89 (1989), 60-82 (in Russian). 
  15. [15] Z. Vakhania, On the ergodic theorems of N. Wiener and D. Ornstein, Soobshch. Akad. Nauk Gruzin. SSR 88 (1977), 281-284 (in Russian). 
  16. [16] Z. Vakhania, On the integrability of the majorant of ergodic means, Trudy Vychisl. Tsentra Akad. Nauk Gruzin. SSR 29 (1990), 43-76 (in Russian). 

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