Diffusive behavior of ergodic sums over rotations

Conze, Jean-Pierre; Isola, Stefano; Borgne, Stéphane Le

doi:10.1142/s0219493719500163

Cited by 6 publications

(5 citation statements)

References 12 publications

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“…In [87] the Erdős-Fortet example was revisited from the perspective of ergodic theory, and was interpreted in terms of the limiting behavior of certain modified ergodic sums, and generalized to cases such as expanding maps, group actions, and chaotic dynamical systems under the assumption of multiple decorrelation. See also [86,88]. The limit distribution of N −1/2 N k=1 cos(2πn k x) for the special sequence n k = k 2 , k ≥ 1, was determined by Jurkat and Van Horne in [141,142,143], and turned out to have finite moments of order < 4, but not of order 4.…”

Section: The Central Limit Theorem For Lacunary Sequencesmentioning

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: The Central Limit Theorem For Lacunary Sequencesmentioning

confidence: 99%

Lacunary sequences in analysis, probability and number theory

Aistleitner¹,

Berkes²,

Tichy³

2023

Preprint

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“…An upper bound for the variance and a lower bound for the mean of the variance are shown in [7]: there are constants 𝐶, 𝑐 > 0 such that…”

Section: Bounds For the Variancementioning

confidence: 99%

“…For an application to the model of rectangular periodic billiard in the plane described in [6], we refer to [7] Using (4.1), we get an upper bound of the distance between the distribution of 𝑋 and the normal law by bounding |E(𝑒 𝑖𝜆𝑋 ) − 𝑒 − 1 2 𝜎 2 𝜆 2 |. We will use the following remarks:…”

Section: Special Values: An Application To the Rectangular Billiard I...mentioning

confidence: 99%

“…For 𝜓 := 1 [0, 1 2 [ − 1 [ 1 2 ,0[ , F. Huveneers [15] studied the existence of a sequence (𝑛 𝑘 ) 𝑛∈N such that (𝜓 𝑛 𝑘 ) after normalization is asymptotically normally distributed. In [7] it was shown that, when 𝛼 has unbounded partial quotients, along some subsequences the ergodic sums of 𝜑 in a class of step functions can be approximated by a Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

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On the CLT for rotations and BV functions

Conze,

Le Borgne

2022

Annales Mathématiques Blaise Pascal

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Let 𝑥 ↦ → 𝑥 + 𝛼 mod 1 be a rotation on the circle and let 𝜑 be a step function. We denote by 𝜑 𝑛 ( 𝑥) the corresponding ergodic sums 𝑛−1 𝑗=0 𝜑 ( 𝑥 + 𝑗 𝛼). For a class of irrational rotations (containing the class with bounded partial quotients) and under a Diophantine condition on the discontinuity points of 𝜑, we show that 𝜑 𝑛 / 𝜑 𝑛 2 is asymptotically Gaussian for 𝑛 in a set of density 1. The proof is based on decorrelation inequalities for the ergodic sums taken at times 𝑞 𝑘 , where (𝑞 𝑘 ) is the sequence of denominators of 𝛼. Another important point is the control of the variance 𝜑 𝑛 2 2 for 𝑛 belonging to a large set of integers. When 𝛼 is a quadratic irrational, the size of this set can be precisely estimated. 0 𝜑(𝑥 + 𝑗 𝛼) over an irrational rotation 𝑥 ↦ → 𝑥 + 𝛼 mod 1. By the Denjoy-Koksma inequality, if 𝜑 is a centered function with bounded variation, the sequence (𝜑 𝑛 ) is uniformly bounded along the sub-sequence of denominators of 𝛼. But, besides, a stochastic behaviour at a certain scale can occur along other sub-sequences (𝑛 𝑘 ). We propose a quantitative analysis of this phenomenon.

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“…We mention that several authors [6,17,28,40] obtained the Central Limit Theorem for circle rotations where normalization is a slowly varying function. However, firstly, the functions considered in those papers are only piecewise smooth and, secondly, there either requires an additional randomness or remove zero density subset of times.…”

Section: Flexibility and Bernoullicitymentioning

confidence: 99%

Flexibility of statistical properties for smooth systems satisfying the central limit theorem

Dolgopyat¹,

Dong²,

Kanigowski³

et al. 2020

Preprint

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In this paper we exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties:• zero entropy;• weak but not strong mixing;• (polynomially) mixing but not K;• K but not Bernoulli;• non Bernoulli and mixing at arbitrary fast polynomial rate. We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index 1.

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Diffusive behavior of ergodic sums over rotations

Cited by 6 publications

References 12 publications

Lacunary sequences in analysis, probability and number theory

Lacunary sequences in analysis, probability and number theory

On the CLT for rotations and BV functions

Flexibility of statistical properties for smooth systems satisfying the central limit theorem

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