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A mean central limit theorem for weakly multiplicative systems and its application to lacunary trigonometric series
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  • Published: June 1991

A mean central limit theorem for weakly multiplicative systems and its application to lacunary trigonometric series

  • Katusi Fukuyama1 

Probability Theory and Related Fields volume 89, pages 159–179 (1991)Cite this article

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Summary

In this paper, we discuss the rate of convergence in the mean central limit theorem for weakly multiplicative systems and apply this result to lacunary trigonometric series under probability measures on ℝ with Hölder continuous distribution function.

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References

  1. Alexits, G.: Sur lá sommabilité des series orthogonales. Acta Math. Acad. Sci. Hung.4, 181–188 (1953)

    Google Scholar 

  2. Alexits, G.: Convergence problems of orthogonal series. Budapest: Académiai Kiadó, 1961

    Google Scholar 

  3. Berkes, I.: On Strassen's version of the log log law for multiplicative systems. Stud. Sci. Math. Hung.8, 425–431 (1973)

    Google Scholar 

  4. Fukyuama, K.: Some limit theorems of almost periodic function systems under the relative measure. J. Math. Kyoto Univ.28, 557–577 (1988)

    Google Scholar 

  5. Fukuyama, K.: Functional central limit theorem and Strassen's law of the iterated logarithms for weakly multiplicative systems. J. Math. Kyoto Univ.30, 625–635 (1990)

    Google Scholar 

  6. Fukuyama, K.: Some limit theorems for weakly multiplicative systems. Colloq. Math. Soc. Janós Bolyai57, 197–214 (1991)

    Google Scholar 

  7. Hausdorff, F.: Dimension and äußeres Maß. Math. Ann.79, 157–179 (1918)

    Google Scholar 

  8. Ibragimov, I.A.: A central limit theorem for a class of dependent random variables. Theory Probab. Appl.8, 83–88 (1963)

    Google Scholar 

  9. Ibragimov, I.A., Linnik, Yu. v.: Independent and stationary sequence of random variables. Groningen, Wolters-Noordhoff Publishing 1971

    Google Scholar 

  10. Kaufman, R.: A problem on lacunary series. Acta Sci. Math.29, 313–316 (1968)

    Google Scholar 

  11. Kershner, R.: On singular Fourier-Stieltjes transforms. Am. J. Math.58, 450–452 (1936)

    Google Scholar 

  12. Kratz, W., Trautner, R.: Zum Gültigkeitbereich des zentralen Grenzwertsatzes und des Gesetzes der großen Zahlen. Acta Math. Acad. Sci. Hung.29, 55–66 (1977)

    Google Scholar 

  13. Móricz, F.: The law of the iterated logarithm and related results for weakly multiplicative systems. Anal. Math.2, 211–229 (1976)

    Google Scholar 

  14. Móricz, F., Révész, P.: Multiplikative rendszerek, (Magyar.) Mat. Lapok28, 43–63 (1980)

    Google Scholar 

  15. Nakata, T.: On the rate of convergence in mean central limit theorem for martingale differences. Rep. Stat. Appl. Res. Union Jap. Sci. Eng.23, 126–131 (1976)

    Google Scholar 

  16. Paditz, L., Šarachmetov, Š.: A mean central limit theorem for multiplicative systems. Math. Nachr.139, 87–94 (1988)

    Google Scholar 

  17. Takahashi, S.: Lacunary trigonometric series and probability. Tôhoku Math. J., II. Ser.22, 502–510 (1970)

    Google Scholar 

  18. Takahashi, S.: On the law of the iterated logarithm for lacunary trigonometric series II. Tôhoku Math. J., II. Ser.27, 391–403 (1975)

    Google Scholar 

  19. Takahashi, S.: Lacunary trigonometric series and some probability measures. Math. Jap.35, 73–77 (1990)

    Google Scholar 

  20. Wiener, N., Wintner, A.: On singular distributions.J. Math. Phys.17, 233–246 (1938)

    Google Scholar 

  21. Zorotarev, V.M.: On asymptotically best constants in refinements of mean central limit theorems. Theory Probab. Appl.9, 268–276 (1964)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, 305, Tsukuba-shi, Ibaraki, Japan

    Katusi Fukuyama

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  1. Katusi Fukuyama
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Cite this article

Fukuyama, K. A mean central limit theorem for weakly multiplicative systems and its application to lacunary trigonometric series. Probab. Th. Rel. Fields 89, 159–179 (1991). https://doi.org/10.1007/BF01366904

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  • Received: 25 April 1990

  • Revised: 23 January 1991

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01366904

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Keywords

  • Distribution Function
  • Stochastic Process
  • Probability Measure
  • Probability Theory
  • Limit Theorem
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