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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10682-8

Published electronically:
February 9, 2011

MathSciNet review:
2784816

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a measurable function satisfying

**1.**Christoph Aistleitner and István Berkes,*On the central limit theorem for 𝑓(𝑛_{𝑘}𝑥)*, Probab. Theory Related Fields**146**(2010), no. 1-2, 267–289. MR**2550364**, https://doi.org/10.1007/s00440-008-0190-6**2.**C. Aistleitner, I. Berkes and R. Tichy, On permutations of Hardy-Littlewood-Pólya 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,*Non-Gaussian limit distributions of lacunary trigonometric series*, Canad. J. Math.**43**(1991), no. 5, 948–959. MR**1138574**, https://doi.org/10.4153/CJM-1991-052-0**5.**I. Berkes and W. Philipp,*An a.s. invariance principle for lacunary series 𝑓(𝑛_{𝑘}𝑥)*, Acta Math. Acad. Sci. Hungar.**34**(1979), no. 1-2, 141–155. MR**546729**, https://doi.org/10.1007/BF01902603**6.**István Berkes and Walter Philipp,*The size of trigonometric and Walsh series and uniform distribution 𝑚𝑜𝑑1*, J. London Math. Soc. (2)**50**(1994), no. 3, 454–464. MR**1299450**, https://doi.org/10.1112/jlms/50.3.454**7.**István Berkes, Walter Philipp, and Robert F. Tichy,*Empirical processes in probabilistic number theory: the LIL for the discrepancy of (𝑛_{𝑘}𝜔)\bmod1*, Illinois J. Math.**50**(2006), no. 1-4, 107–145. MR**2247826****8.**P. Erdős,*On trigonometric sums with gaps*, Magyar Tud. Akad. Mat. Kutató Int. Közl**7**(1962), 37–42 (English, with Russian summary). MR**0145264****9.**K. Fukuyama,*The law of the iterated logarithm for the discrepancies of a permutation of {𝑛_{𝑘}𝑥}*, Acta Math. Hungar.**123**(2009), no. 1-2, 121–125. MR**2496484**, https://doi.org/10.1007/s10474-008-8067-9**10.**I. S. Gál,*A theorem concerning Diophantine approximations*, Nieuw Arch. Wiskunde (2)**23**(1949), 13–38. MR**0027788****11.**V. F. Gapoškin,*Lacunary series and independent functions*, Uspehi Mat. Nauk**21**(1966), no. 6 (132), 3–82 (Russian). MR**0206556****12.**V. F. Gaposhkin, The central limit theorem for some weakly dependent sequences.*Theory Probab. Appl.***15**(1970), 649-666.**13.**Shin-ichi Izumi,*Notes on Fourier analysis. XLIV. On the law of the iterated logarithm of some sequences of functions*, J. Math. Tokyo**1**(1951), 1–22. MR**0051962****14.**M. Kac,*On the distribution of values of sums of the type ∑𝑓(2^{𝑘}𝑡)*, Ann. of Math. (2)**47**(1946), 33–49. MR**0015548**, https://doi.org/10.2307/1969033**15.**M. Kac,*Probability methods in some problems of analysis and number theory*, Bull. Amer. Math. Soc.**55**(1949), 641–665. MR**0031504**, https://doi.org/10.1090/S0002-9904-1949-09242-X**16.**J. F. Koksma,*On a certain integral in the theory of uniform distribution*, Nederl. Akad. Wetensch., Proc. Ser. A. 54 = Indagationes Math.**13**(1951), 285–287. MR**0045165****17.**Gisiro Maruyama,*On an asymptotic property of a gap sequence*, Kōdai Math. Sem. Rep.**2**(1950), 31–32. {Volume numbers not printed on issues until Vol. 7 (1955).}. MR**0038470****18.**Shigeru Takahashi,*On lacunary trigonometric series*, Proc. Japan Acad.**41**(1965), 503–506. MR**0196377****19.**Shigeru Takahashi,*On the law of the iterated logarithm for lacunary trigonometric series*, Tôhoku Math. J. (2)**24**(1972), 319–329. Collection of articles dedicated to Gen-ichirô Sunouchi on his sixtieth birthday. MR**0330905**, https://doi.org/10.2748/tmj/1178241542**20.**A. Zygmund,*Trigonometric series. Vol. I, II*, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. With a foreword by Robert A. Fefferman. MR**1963498**

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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:
https://doi.org/10.1090/S0002-9939-2011-10682-8

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.