Critical Lil Behavior of the Trigonometric System

Berkes, I.

doi:10.2307/2154417

Cited by 4 publications

(1 citation statement)

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“…Broadly speaking, the "almost independent" behavior of sums of dilated functions breaks down when the lacunarity condition is relaxed. Many papers have been devoted to this effect; see in particular [57,59,102,184]. In order to maintain the "almost independent" behavior of the sum, there are two natural routes to take.…”

Section: Arithmetic Effects: Diophantine Equations and Sums Of Common...mentioning

confidence: 99%

Lacunary sequences in analysis, probability and number theory

Aistleitner¹,

Berkes²,

Tichy³

2023

Preprint

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In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.Contents 24 8. The subsequence principle 28 9. New results: Exact criteria for the central limit theorem for subsequences 36 Acknowledgments 49 References 50

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Section: Arithmetic Effects: Diophantine Equations and Sums Of Common...mentioning

confidence: 99%

Lacunary sequences in analysis, probability and number theory

Aistleitner¹,

Berkes²,

Tichy³

2023

Preprint

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On the Law of The Iterated Logarithm for Rapidly Increasing Subsequences

Schwabe

Gut

1996

Mathematische Nachrichten

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The usual law of the iterated logarithm states that the partial sums Sn of independent and identically distributed random variables can be normalized by the sequence an = d -, such that limsup,,, &/a, = t/z a. 9.. As has been pointed out by GUT (1986) the law fails if one considers the limsup along subsequences which increase faster than exponentially. In particular, for very rapidly increasing subsequences {nk, k 2 1) one has limsupk-oo S,, /ank = 0 a. 9.. In these cases the normalizing constants ank have to be replaced by J;6;i.pk to obtain a non-trivial limiting behaviour: limsupk4ao S n k / d a = 0 a. 8. . We will present an intelligible argument for this structural change and apply it to related results. lishes the convergence of the mean; $Sn t 0 almost surely (a.s.). The weights 1/n lead to a degenerate limiting law and hence tend to zero too fast. On the other hand, the norming factors l / f i produce the Central Limit Theorem; h S n 3 N(0, l), but S,/J;;; is a. s. diverging. Thus the rate of convergence of the mean S,/n is slower than AMS subject classification: 60F15, 60G50 Key words: Sums of i.i.d. random variables, law of the iterated logarithm, subsequence, geometric increase, Chung's law of the iterated logarithm, Kolmogorov statistic.(in Russian).

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An optimal condition for the LIL for trigonometric series

Berkes¹

1995

Trans. Amer. Math. Soc.

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Abstract.By a classical theorem , Erdos-Gal [3]), if (n*.) is a sequence of positive integers satisfying H/t+i/n/fc > q > 1 (A: =1,2,...) then (cos^x) obeys the law of the iterated logarithm, i.e., (1) limsup(JVloglogAr)~1^2 ^ cos«^x= 1 a.e. N^°° k q > 1 can be essentially weakened here but the problem of finding the precise gap condition for the LIL (1) has remained open. In this paper we find, using combinatorial methods, an optimal gap condition for the upper half of the LIL, i.e., the inequality < 1 in (1).

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Critical Lil Behavior of the Trigonometric System

Cited by 4 publications

References 4 publications

Lacunary sequences in analysis, probability and number theory

Lacunary sequences in analysis, probability and number theory

On the Law of The Iterated Logarithm for Rapidly Increasing Subsequences

An optimal condition for the LIL for trigonometric series

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