Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the law of the iterated logarithm for the discrepancy of lacunary sequences II

Author: Christoph Aistleitner
Journal: Trans. Amer. Math. Soc. 365 (2013), 3713-3728
MSC (2010): Primary 11K38, 60F15, 11D04, 11J83, 42A55
Published electronically: October 31, 2012
MathSciNet review: 3042600
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By a classical heuristics, lacunary function systems exhibit many asymptotic properties which are typical for systems of independent random variables. For example, for a large class of functions $ f$ the system $ (f(n_k x))_{k \geq 1}$, where $ (n_k)_{k \geq 1}$ is a lacunary sequence of integers, satisfies a law of the iterated logarithm (LIL) of the form

$\displaystyle c_1 \leq \limsup _{N \to \infty } \frac {\sum _{k=1}^N f(n_k x)}{\sqrt {2 N \log \log N}} \leq c_2 \qquad \textup {a.e.},$ (1)

where $ c_1,c_2$ are appropriate positive constants. In a previous paper we gave a criterion, formulated in terms of the number of solutions of certain linear Diophantine equations, which guarantees that the value of the $ \limsup $ in (1) equals the $ L^2$-norm of $ f$ for $ \textup {a.e.}$ $ x$, which is exactly what one would also expect in the case of i.i.d. random variables. This result can be used to prove a precise LIL for the discrepancy of $ (n_k x)_{k \geq 1}$, which corresponds to the Chung-Smirnov LIL for the Kolmogorov-Smirnov-statistic of i.i.d. random variables.

In the present paper we give a full solution of the problem in the case of ``stationary'' Diophantine behavior, by this means providing a unifying explanation of the aforementioned ``regular'' LIL behavior and the ``irregular'' LIL behavior which has been observed by Kac, Erdős, Fortet and others.

References [Enhancements On Off] (What's this?)

  • 1. C. Aistleitner.
    Irregular discrepancy behavior of lacunary series.
    Monatsh. Math., 160(1):1-29, 2010. MR 2610309 (2011i:11117)
  • 2. C. Aistleitner.
    Irregular discrepancy behavior of lacunary series II.
    Monatsh. Math., 161(3):255-270, 2010. MR 2726213
  • 3. C. Aistleitner.
    On the class of limits of lacunary trigonometric series.
    Acta Math. Hungar., 129(1-2):1-23, 2010. MR 2725832 (2011i:42011)
  • 4. C. Aistleitner.
    On the law of the iterated logarithm for the discrepancy of lacunary sequences.
    Trans. Amer. Math. Soc., 362(11):5967-5982, 2010. MR 2661504
  • 5. C. Aistleitner and I. Berkes.
    On the central limit theorem for $ f(n_kx)$.
    Probab. Theory Related Fields, 146(1-2):267-289, 2010. MR 2550364 (2010i:42015)
  • 6. C. Aistleitner and I. Berkes.
    Probability and metric discrepancy theory.
    Stoch. Dyn., 11(1):183-207, 2011. MR 2771348
  • 7. I. Berkes and W. Philipp.
    An a.s. invariance principle for lacunary series $ f(n_{k}x)$.
    Acta Math. Acad. Sci. Hungar., 34(1-2):141-155, 1979. MR 546729 (80i:60042)
  • 8. J.-P. Conze and S. Le Borgne.
    Limit law for some modified ergodic sums.
    Stoch. Dyn., 11(1):107-133, 2011. MR 2771345
  • 9. M. Drmota and R. F. Tichy.
    Sequences, discrepancies and applications, volume 1651 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 1997. MR 1470456 (98j:11057)
  • 10. P. Erdös and I. S. Gál.
    On the law of the iterated logarithm. I, II.
    Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math., 17:65-76, 77-84, 1955. MR 0069309 (16:1016g)
  • 11. R. Fortet.
    Sur une suite egalement répartie.
    Studia Math., 9:54-70, 1940. MR 0005546 (3:169e)
  • 12. K. Fukuyama.
    The law of the iterated logarithm for discrepancies of $ \{\theta ^nx\}$.
    Acta Math. Hungar., 118(1-2):155-170, 2008. MR 2378547 (2008m:60049)
  • 13. K. Fukuyama.
    A central limit theorem and a metric discrepancy result for sequences with bounded gaps.
    In Dependence in probability, analysis and number theory, pages 233-246. Kendrick Press, Heber City, UT, 2010. MR 2731069
  • 14. K. Fukuyama and S. Miyamoto.
    Metric discrepancy results for Erdős-Fortet sequence.
    Studia Sci. Math. Hung.
  • 15. V. F. Gapoškin.
    Lacunary series and independent functions.
    Uspehi Mat. Nauk, 21(6 (132)):3-82, 1966. MR 0206556 (34:6374)
  • 16. M. Kac.
    On the distribution of values of sums of the type $ \sum f(2^k t)$.
    Ann. of Math. (2), 47:33-49, 1946. MR 0015548 (7:436f)
  • 17. M. Kac.
    Probability methods in some problems of analysis and number theory.
    Bull. Amer. Math. Soc., 55:641-665, 1949. MR 0031504 (11:161b)
  • 18. L. Kuipers and H. Niederreiter.
    Uniform distribution of sequences.
    Wiley-Interscience [John Wiley & Sons], New York, 1974.
    Pure and Applied Mathematics. MR 0419394 (54:7415)
  • 19. G. Maruyama.
    On an asymptotic property of a gap sequence.
    Kōdai Math. Sem. Rep., 2:31-32, 1950.
    {Volume numbers not printed on issues until Vol. 7 (1955).} MR 0038470 (12:406e)
  • 20. W. Philipp.
    Limit theorems for lacunary series and uniform distribution $ {\rm mod}\ 1$.
    Acta Arith., 26(3):241-251, 1974/75. MR 0379420 (52:325)
  • 21. G. R. Shorack and J. A. Wellner.
    Empirical processes with applications to statistics.
    Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. MR 838963 (88e:60002)
  • 22. A. Zygmund.
    Trigonometric series. Vol. I, II.
    Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988.
    Reprint of the 1979 edition. MR 933759 (89c:42001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11K38, 60F15, 11D04, 11J83, 42A55

Retrieve articles in all journals with MSC (2010): 11K38, 60F15, 11D04, 11J83, 42A55

Additional Information

Christoph Aistleitner
Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

Keywords: Discrepancy, law of the iterated logarithm, Diophantine equations, lacunary series
Received by editor(s): July 21, 2011
Received by editor(s) in revised form: November 1, 2011
Published electronically: October 31, 2012
Additional Notes: The author’s research was supported by the Austrian Research Foundation (FWF), Project S9603-N23.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society