Central Limit Theorem and Diophantine Approximations

Sergey G. Bobkov

doi:10.1007/s10959-017-0770-4

Central Limit Theorem and Diophantine Approximations

Sergey G. Bobkov

School of Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Let F_n denote the distribution function of the normalized sum Zn=(X1+⋯+Xn)/(σn) of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of F_n to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of F_n by the Edgeworth corrections (modulo logarithmically growing factors in n), are given in terms of the characteristic function of X₁. Particular cases of the problem are discussed in connection with Diophantine approximations.

Original language	English (US)
Pages (from-to)	2390-2411
Number of pages	22
Journal	Journal of Theoretical Probability
Volume	31
Issue number	4
DOIs	https://doi.org/10.1007/s10959-017-0770-4
State	Published - Dec 1 2018

Bibliographical note

Funding Information:
Partially supported by the NSF Grant DMS-1612961.

Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.

Keywords

Central limit theorem
Diophantine approximation
Edgeworth expansions

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10.1007/s10959-017-0770-4

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abstract = "Let Fn denote the distribution function of the normalized sum Zn=(X1+⋯+Xn)/(σn) of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of Fn to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of Fn by the Edgeworth corrections (modulo logarithmically growing factors in n), are given in terms of the characteristic function of X1. Particular cases of the problem are discussed in connection with Diophantine approximations.",

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N2 - Let Fn denote the distribution function of the normalized sum Zn=(X1+⋯+Xn)/(σn) of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of Fn to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of Fn by the Edgeworth corrections (modulo logarithmically growing factors in n), are given in terms of the characteristic function of X1. Particular cases of the problem are discussed in connection with Diophantine approximations.

AB - Let Fn denote the distribution function of the normalized sum Zn=(X1+⋯+Xn)/(σn) of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of Fn to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of Fn by the Edgeworth corrections (modulo logarithmically growing factors in n), are given in terms of the characteristic function of X1. Particular cases of the problem are discussed in connection with Diophantine approximations.

KW - Central limit theorem

KW - Diophantine approximation

KW - Edgeworth expansions

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JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 4

ER -

Central Limit Theorem and Diophantine Approximations

Abstract

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