## The Annals of Statistics

### Khinchine’s theorem and Edgeworth approximations for weighted sums

Sergey G. Bobkov

#### Abstract

Let $F_{n}$ denote the distribution function of the normalized sum of $n$ i.i.d. random variables. In this paper, polynomial rates of approximation of $F_{n}$ by the corrected normal laws are considered in the model where the underlying distribution has a convolution structure. As a basic tool, the convergence part of Khinchine’s theorem in metric theory of Diophantine approximations is extended to the class of product characteristic functions.

#### Article information

Source
Ann. Statist., Volume 47, Number 3 (2019), 1616-1633.

Dates
Revised: May 2018
First available in Project Euclid: 13 February 2019

https://projecteuclid.org/euclid.aos/1550026851

Digital Object Identifier
doi:10.1214/18-AOS1728

Mathematical Reviews number (MathSciNet)
MR3911124

Zentralblatt MATH identifier
07053520

Subjects
Primary: 60F05: Central limit and other weak theorems

#### Citation

Bobkov, Sergey G. Khinchine’s theorem and Edgeworth approximations for weighted sums. Ann. Statist. 47 (2019), no. 3, 1616--1633. doi:10.1214/18-AOS1728. https://projecteuclid.org/euclid.aos/1550026851

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