Abstract
Abstract We prove Strassen’s law of the iterated logarithm for sums \({\sum^{N}_{k=1} f(n_kx),}\) where f is a smooth periodic function on the real line and \({(n_k)_{k \geqq 1}}\) is an increasing random sequence. Our results show that classical results of the theory of lacunary series remain valid for sequences with random gaps, even in the nonharmonic case and if the Hadamard gap condition fails.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aistleitner C, Fukuyama K.: On the law of the iterated logarithm for trigonometric series with bounded gaps. Probab. Theory Related Fields, 154, 607–620 (2012)
Berkes I.: On the central limit theorem for lacunary trigonometric series, Analysis Math.. , 4, 159–180 (1978)
Berkes I.: A central limit theorem for trigonometric series with small gaps, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47, 157–161 (1979)
I. Berkes and M. Weber, On the convergence of \({\sum c_kf(n_kx),}\) Mem. Amer. Math. Soc.,201 (2009), no. 943, viii+72 pp.
Chover J.: On Strassen’s version of the loglog law, Z. Wahrsch. Verw. Gebiete, 8, 83–90 (1967)
Erdős P.: On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kut. Int. Közl., 7, 37–42 (1962)
Erdős P, Gál I. S.: On the law of the iterated logarithm. Proc. Nederl. Akad. Wetensch. Ser A, 58, 65–84 (1955)
K. Fukuyama, A central limit theorem and a metric discrepancy result for sequences with bounded gaps, in: Dependence in Probability, Analysis and Number Theory, Kendrick Press (2010), pp. 233–246.
Fukuyama K.: A central limit theorem for trigonometric series with bounded gaps, Prob. Theory Rel. Fields, 149, 139–148 (2011)
Gaposhkin V. F.: Lacunary series and independent functions, Russian Math. Surveys 21, 1–82 (1966)
H. Halberstam and K. F. Roth, Sequences, Vol. I., Clarendon Press (Oxford, 1966).
Kac M.: Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc., 55, 641–665 (1949)
Major P.: A note on Kolmogorov’s law of iterated logarithm, Studia Sci. Math. Hungar., 12, 161–167 (1977)
Salem R, Zygmund A.: On lacunary trigonometric series, Proc. Nat. Acad. Sci. USA, 33, 333–338 (1947)
Schatte P.: On the asymptotic uniform distribution of sums reduced mod 1, Math. Nachr., 115(275–281), 115 275–281 (1984)
Schatte P.: On a law of the iterated logarithm for sums mod 1 with applications to Benford’s law, Prob. Th. Rel. Fields, 77, 167–178 (1988)
Strassen V.: An invariance principle for the law of the iterated logarithm, Z. Wahrsch. Verw. Gebiete, 3(211–226), 3 211–226 (1964)
Takahashi S.: On the law of the iterated logarithm for lacunary trigonometric series, Tohoku Math. J., 24, 319–329 (1972)
Weber M.: Discrepancy of randomly sampled sequences of integers, Math. Nachr., 271, 105–110 (2004)
Weiss M.: On the law of the iterated logarithm for uniformly bounded orthonormal systems, Trans. Amer. Math. Soc., 92, 531–553 (1959)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by FWF Projekt W1230.
Rights and permissions
About this article
Cite this article
Raseta, M. On Lacunary Series with Random Gaps. Acta Math. Hungar. 144, 150–161 (2014). https://doi.org/10.1007/s10474-014-0430-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-014-0430-4