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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the asymptotic behavior of weakly lacunary series


Authors: C. Aistleitner, I. Berkes and R. Tichy
Journal: Proc. Amer. Math. Soc. 139 (2011), 2505-2517
MSC (2010): Primary 42A55, 42A61, 11D04, 60F05, 60F15
Published electronically: February 9, 2011
MathSciNet review: 2784816
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Abstract: Let $ f$ be a measurable function satisfying

$\displaystyle f(x+1)=f(x), \quad \int_0^1 f(x) dx=0, \quad \textrm{Var}_{[0,1]} f < + \infty, $

and let $ (n_k)_{k\ge 1}$ be a sequence of integers satisfying $ n_{k+1}/n_k \ge q >1$ $ (k=1, 2, \ldots)$. By the classical theory of lacunary series, under suitable Diophantine conditions on $ n_k$, $ (f(n_kx))_{k\ge 1}$ satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences $ (n_k)_{k\ge 1}$ as well, but as Fukuyama showed, the behavior of $ f(n_kx)$ is generally not permutation-invariant; 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 permutation-invariant CLT and LIL without any growth conditions on $ (n_k)_{k\ge 1}$ and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if $ f(x)=\sin 2\pi x$ and $ (n_k)_{k\ge 1}$ grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences $ (n_k)_{k\ge 1}$ growing faster than polynomially, $ (f(n_kx))_{k\ge 1}$ has permutation-invariant behavior.


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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/S0002-9939-2011-10682-8
PII: S 0002-9939(2011)10682-8
Keywords: Lacunary series, central limit theorem, law of the iterated logarithm, permutation-invariance, 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 S9603-N23.
The second author’s research was supported by FWF grant S9603-N23 and OTKA grants K 67961 and K 81928.
The third author’s research was supported by FWF grant S9603-N23.
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.