Central Limit Theorem and Diophantine Approximations

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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.

Original languageEnglish (US)
Pages (from-to)2390-2411
Number of pages22
JournalJournal of Theoretical Probability
Volume31
Issue number4
DOIs
StatePublished - Dec 1 2018

Bibliographical note

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

Keywords

  • Central limit theorem
  • Diophantine approximation
  • Edgeworth expansions

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