On the law of the iterated logarithm for the discrepancy of lacunary sequences II
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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 \begin{equation}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 \mathrm {a.e.}, \end{equation} 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 $\mathrm {a.e.}$ $x$, 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.
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Additional Information
- Christoph Aistleitner
- Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
- Email: aistleitner@math.tugraz.at
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3713-3728
- MSC (2010): Primary 11K38, 60F15, 11D04, 11J83, 42A55
- DOI: https://doi.org/10.1090/S0002-9947-2012-05740-0
- MathSciNet review: 3042600