On the asymptotic behavior of weakly lacunary series
Authors:
C. Aistleitner, I. Berkes and R. Tichy
Journal:
Proc. Amer. Math. Soc. 139 (2011), 25052517
MSC (2010):
Primary 42A55, 42A61, 11D04, 60F05, 60F15
Published electronically:
February 9, 2011
MathSciNet review:
2784816
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Abstract: Let be a measurable function satisfying and let be a sequence of integers satisfying . By the classical theory of lacunary series, under suitable Diophantine conditions on , satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences as well, but as Fukuyama showed, the behavior of is generally not permutationinvariant; e.g. a rearrangement of the sequence can ruin the CLT and LIL. In this paper we construct an infinite order Diophantine condition implying the permutationinvariant CLT and LIL without any growth conditions on and show that the known finite order Diophantine conditions in the theory do not imply permutationinvariance even if and grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences growing faster than polynomially, has permutationinvariant behavior.
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 2.
 C. Aistleitner, I. Berkes and R. Tichy, On permutations of HardyLittlewoodPólya sequences. Transactions of the AMS, to appear.
 3.
 C. Aistleitner, I. Berkes and R.F. Tichy, Lacunarity, symmetry and Diophantine equations. Preprint.
 4.
 I. Berkes, NonGaussian limit distributions of lacunary trigonometric series. Canad. J. Math. 43 (1991), 948959. MR 1138574 (92k:60108)
 5.
 I. Berkes and W. Philipp, An a.s. invariance principle for lacunary series . Acta Math. Acad. Sci. Hung. 34 (1979), 141155. MR 546729 (80i:60042)
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 I. Berkes and W. Philipp, The size of trigonometric and Walsh series and uniform distribution mod 1. J. Lond. Math. Soc. 50 (1994), 454464. MR 1299450 (96e:11099)
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 I. Berkes, W. Philipp and R.F. Tichy, Empirical processes in probabilistic number theory: the LIL for the discrepancy of mod 1. Illinois J. Math. 50 (2006), 107145. MR 2247826 (2008a:60064)
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 P. Erdős, On trigonometric sums with gaps. Magyar Tud. Akad. Mat. Kut. Int. Közl. 7 (1962), 3742. MR 0145264 (26:2797)
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 I.S. Gál, A theorem concerning Diophantine approximations. Nieuw. Arch. Wiskunde (2) 23 (1949), 1338. MR 0027788 (10:355a)
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Additional Information
C. Aistleitner
Affiliation:
Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Email:
aistleitner@math.tugraz.at
I. Berkes
Affiliation:
Institute of Statistics, Graz University of Technology, Münzgrabenstraße 11, 8010 Graz, Austria
Email:
berkes@tugraz.at
R. Tichy
Affiliation:
Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Email:
tichy@tugraz.at
DOI:
http://dx.doi.org/10.1090/S000299392011106828
Keywords:
Lacunary series,
central limit theorem,
law of the iterated logarithm,
permutationinvariance,
Diophantine equations
Received by editor(s):
May 16, 2010
Received by editor(s) in revised form:
July 4, 2010
Published electronically:
February 9, 2011
Additional Notes:
The first author’s research was supported by FWF grant S9603N23.
The second author’s research was supported by FWF grant S9603N23 and OTKA grants K 67961 and K 81928.
The third author’s research was supported by FWF grant S9603N23.
Communicated by:
Richard C. Bradley
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
