## The Annals of Statistics

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

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

#### 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**

Received: October 2017

Revised: May 2018

First available in Project Euclid: 13 February 2019

**Permanent link to this document**

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

**Keywords**

Central limit theorem Edgeworth approximations

#### 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