View More View Less
  • 1 Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
Restricted access

Abstract

This paper is mainly concerned with the limit distribution of on the unit interval when the increasing sequence {nk} has bounded gaps, i.e., 1≤nk+1nk=O(1). By Bobkov–Götze [4], it was proved that the limiting variance must be less than 1/2 in this case. They proved that the centered Gaussian distribution with variance 1/4 together with mixtures of Gaussian distributions belonging to a huge class can be limit distributions. In this paper it is proved that any Gaussian distribution with variance less than 1/2 can be a limit distribution.

  • [1] Berkes, I. 1978 On the central limit theorem for lacunary trigonometric series Anal. Math. 4 159180 .

  • [2] Berkes, I. 1979 A central limit theorem for trigonometric series with small gaps Z. Wahr. verw. Geb. 47 157161 .

  • [3] Billingsley, P. 1995 Probability and Measure 3 Wiley New York.

  • [4] Bobkov, S., Götze, F. 2007 Concentration inequalities and limit theorems for randomized sums Prob. Theory Related Fields 137 4981 .

  • [5] Erdös, P. 1962 On trigonometric series with gaps Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 3742.

  • [6] Fukuyama, K., A central limit theorem for trigonometric series with bounded gaps, Prob. Theory Related Fields, to appear.

  • [7] Itô, K. 1984 Introduction to Probability Theory Cambridge Univ. Press Cambridge.

  • [8] Kac, M. 1939 Note on power series with big gaps Amer. J. Math. 61 473476 .

  • [9] Loève, M. 1977 Probability Theory I 4 Springer Berlin.

  • [10] Salem, R., Zygmund, A. 1947 On lacunary trigonometric series Proc. Nat. Acad. Sci. 33 333338 .

  • [11] Salem, R., Zygmund, A. 1954 Some properties of trigonometric series whose terms have random signs Acta. Math. 91 245301 .

  • [12] Takahashi, S. 1965 On trigonometric series with gaps Tôhoku Math. J. 17 227234 .

  • [13] Takahashi, S. 1968 On lacunary trigonometric series II Proc. Japan Acad. 44 766770 .

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Jan 2021 0 0 0
Feb 2021 0 0 0
Mar 2021 1 0 0
Apr 2021 0 0 0
May 2021 0 0 0
Jun 2021 0 0 0
Jul 2021 0 0 0