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1992 | 103 | 1 | 1-15
Tytuł artykułu

On the distribution function of the majorant of ergodic means

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
103
Numer
1
Strony
1-15
Opis fizyczny
Daty
wydano
1992
otrzymano
1989-07-19
poprawiono
1991-07-05
Twórcy
  • A. Razmadze Mathematical Institute, Georgian Academy of Sciences, Rukhadze 1, 380093 Tbilisi, Georgia
Bibliografia
  • [1] P. Billingsley, Ergodic Theory and Information, Wiley, 1965.
  • [2] B. Davis, On the integrability of the ergodic maximal function, Studia Math. 73 (1982), 153-167.
  • [3] B. Davis, Stopping rules for $S_n/n$, and the class L log L, Z. Warsch. Verw. Gebiete 17 (1971), 147-150.
  • [4] J. L. Doob, Stochastic Processes, Wiley, 1953.
  • [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] A. M. Garsia, A simple proof of E. Hopf's maximal ergodic theorem, J. Math. Mech. 14 (1965), 381-382.
  • [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.
  • [8] B. J. McCabe and L. A. Shepp, On the supremum of $S_n/n$, Ann. Math. Statist. 41 (1970), 2166-2168.
  • [9] J. Neveu, The filling scheme and the Chacon-Ornstein theorem, Israel J. Math. 33 (1979), 368-377.
  • [10] D. Ornstein, A remark on the Birkhoff ergodic theorem, Illinois J. Math. 15 (1971), 77-79.
  • [11] K. E. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983.
  • [12] F. Riesz, Sur la théorie ergodique, Comment. Math. Helv. 17 (1944/45), 221-239.
  • [13] R. Sato, On the ratio maximal function for an ergodic flow, Studia Math. 80 (1984), 129-139.
  • [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] Z. Vakhania, On the ergodic theorems of N. Wiener and D. Ornstein, Soobshch. Akad. Nauk Gruzin. SSR 88 (1977), 281-284 (in Russian).
  • [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).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv103i1p1bwm
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