# 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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## 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 $f*=su{p}_{N}1/N{\sum }_{m=0}^{N-1}f\circ {T}^{m}$. In this paper we mainly investigate the question of whether (i) ${ʃ}_{a}^{\infty }|\mu \left(f*>t\right)-1/t{ʃ}_{\left(f*>t\right)}fd\mu |dt<\infty$ and whether (ii) ${ʃ}_{a}^{\infty }|\mu \left(f*>t\right)-1/t{ʃ}_{\left(f>t\right)}fd\mu |dt<\infty$ 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.

## 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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1. [1] P. Billingsley, Ergodic Theory and Information, Wiley, 1965. Zbl0141.16702
2. [2] B. Davis, On the integrability of the ergodic maximal function, Studia Math. 73 (1982), 153-167. Zbl0496.28017
3. [3] B. Davis, Stopping rules for ${S}_{n}/n$, and the class L log L, Z. Warsch. Verw. Gebiete 17 (1971), 147-150. Zbl0194.50102
4. [4] J. L. Doob, Stochastic Processes, Wiley, 1953.
5. [5] L. Epremidze, On the distribution function of the majorant of ergodic means, Seminar Inst. Prikl. Mat. Tbilis. Univ. 3 (2) (1988), 89-92 (in Russian).
6. [6] A. M. Garsia, A simple proof of E. Hopf's maximal ergodic theorem, J. Math. Mech. 14 (1965), 381-382. Zbl0178.38601
7. [7] R. L. Jones, New proofs for the maximal ergodic theorem and the Hardy-Littlewood maximal theorem, Proc. Amer. Math. Soc. 87 (1983), 681-684. Zbl0551.28018
8. [8] B. J. McCabe and L. A. Shepp, On the supremum of ${S}_{n}/n$, Ann. Math. Statist. 41 (1970), 2166-2168. Zbl0226.60067
9. [9] J. Neveu, The filling scheme and the Chacon-Ornstein theorem, Israel J. Math. 33 (1979), 368-377. Zbl0428.28011
10. [10] D. Ornstein, A remark on the Birkhoff ergodic theorem, Illinois J. Math. 15 (1971), 77-79. Zbl0212.40102
11. [11] K. E. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983.
12. [12] F. Riesz, Sur la théorie ergodique, Comment. Math. Helv. 17 (1944/45), 221-239. Zbl0063.06500
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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