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A lower bound in the tail law of the iterated logarithm for lacunary trigonometric series


Authors: Santosh Ghimire and Charles N. Moore
Journal: Proc. Amer. Math. Soc. 142 (2014), 3207-3216
MSC (2010): Primary 42A55; Secondary 60F15
DOI: https://doi.org/10.1090/S0002-9939-2014-12055-7
Published electronically: June 2, 2014
MathSciNet review: 3223376
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Abstract | References | Similar Articles | Additional Information

Abstract: Salem and Zygmund obtained an upper bound for a tail law of the iterated logarithm for sums of the form $ \sum _{k=N}^{\infty } a_{k} \cos (n_{k}x)+b_k \sin (n_{k}x)$, where $ n_{k}$ satisfies a Hadamard gap condition and $ \sum _{k=1}^{\infty } a_k^2 + b_k^2 < \infty .$ Here we obtain a lower bound in their result under the same hypotheses.


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  • [1] Rodrigo Bañuelos and Charles N. Moore, Probabilistic behavior of harmonic functions, Progress in Mathematics, vol. 175, Birkhäuser Verlag, Basel, 1999. MR 1707297 (2001j:31003)
  • [2] Kai Lai Chung, A course in probability theory, 2nd ed., Probability and Mathematical Statistics, Vol. 21. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0346858 (49 #11579)
  • [3] P. Erdős and I. S. Gál, On the law of the iterated logarithm. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 58 (1955), 65-84.MR 0069309
  • [4] V. F. Gaposhkin, On the speed of convergence to the Gaussian law of weighted sums of gap series, Theory of Probability and its Applications 13 (1968), 421-438.
  • [5] A. Khintchine, Über einen Satz der Wahrscheinlichkeitsrechnung, Fundamenta Mathematica, 6 (1924), 9-20.
  • [6] N. Kolmogorov, Über des Gesetz des iterierten Logarithmus, Mathematische Annalen 101 (1929), 126-139.
  • [7] R. Salem and A. Zygmund, La loi du logarithme itéré pour les séries trigonométriques lacunaires, Bull. Sci. Math. (2) 74 (1950), 209-224 (French). MR 0039828 (12,605c)
  • [8] Mary Weiss, The law of the iterated logarithm for lacunary trigonometric series, Trans. Amer. Math. Soc. 91 (1959), 444-469. MR 0108681 (21 #7396)
  • [9] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776 (21 #6498)

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Additional Information

Santosh Ghimire
Affiliation: Department of Science and Humanities, Tribhuvan University, Pulchowk Campus, Lalitpur, Kathmandu, Nepal
Email: ghimire@math.ksu.edu

Charles N. Moore
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Address at time of publication: Department of Mathematics, Washington State University, Pullman, Washington 99164
Email: cnmoore@math.wsu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12055-7
Keywords: Law of the iterated logarithm, lacunary trigonometric series
Received by editor(s): May 21, 2012
Received by editor(s) in revised form: October 8, 2012
Published electronically: June 2, 2014
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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