Abstract
In 1975 Philipp showed that for any increasing sequence (n k ) of positive integers satisfying the Hadamard gap condition n k+1/n k > q > 1, k ≥ 1, the discrepancy D N of (n k x) mod 1 satisfies the law of the iterated logarithm
Recently, Fukuyama computed the value of the lim sup for sequences of the form n k = θk, θ > 1, and in a preceding paper the author gave a Diophantine condition on (n k ) for the value of the limsup to be equal to 1/2, the value obtained in the case of i.i.d. sequences. In this paper we utilize this number-theoretic connection to construct a lacunary sequence (n k ) for which the lim sup in the LIL for the star-discrepancy \({D_N^*}\) is not a constant a.e. and is not equal to the lim sup in the LIL for the discrepancy D N .
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Aistleitner, C.: On the law of the iterated logarithm for the discrepancy of lacunary sequences. Trans. Am. Math. Soc. (2008, to appear)
Aistleitner, C., Berkes, I.: On the central limit theorem for f(n k x). Prob. Theory Related Fields (2008, to appear)
Berkes I.: On the central limit theorem for lacunary trigonometric series. Anal. Math. 4, 159–180 (1978)
Berkes I., Philipp W.: An a.s. invariance principle for lacunary series f(n k x). Acta Math. Acad. Sci. Hungar. 34, 141–155 (1979)
Fukuyama, K.: A law of the iterated logarithm for discrepancies: non-constant limsup. Monatsh. Math. (2008, to appear)
Fukuyama K.: The law of the iterated logarithm for discrepancies of {θn x }. Acta Math. Hung. 118, 155–170 (2008)
Kac M.: Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc. 55, z641–665 (1949)
Kesten H.: The discrepancy of random sequences {kx}. Acta Arith. 10, 183–213 (1964/1965)
Khinchin A.: Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92, 115–125 (1924)
Philipp W.: Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26, 241–251 (1975)
Shorack R., Wellner J.: Empirical Processes with Applications to Statistics. Wiley, New York (1986)
Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. In: Proceedings of Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), vol. II. Contributions to Probability Theory, pp. 315–343 (1967)
Takahashi S.: An asymptotic property of a gap sequence. Proc. Jpn. Acad. 38, 101–104 (1962)
Weyl H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)
Zygmund, A.: Trigonometric Series, vol. I, II. Reprint of the 1979 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988)
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Communicated by J. Schoißengeier.
This research was supported by the Austrian Research Foundation (FWF), Project S9603-N13.
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Aistleitner, C. Irregular discrepancy behavior of lacunary series. Monatsh Math 160, 1–29 (2010). https://doi.org/10.1007/s00605-008-0067-x
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DOI: https://doi.org/10.1007/s00605-008-0067-x