## The Annals of Probability

- Ann. Probab.
- Volume 27, Number 2 (1999), 1072-1098.

### Asymptotic Distribution of Quadratic Forms

#### Abstract

We consider quadratic forms

Q_n = \sum_{1 \le j \neq k \le n} a_{jk}X_j X_k,

where $X_j$ are i.i.d. random variables with finite third moment. We obtain optimal bounds for the Kolmogorov distance between the distribution of $Q_n$ and the distribution of the same quadratic forms with $X_j$ replaced by corresponding Gaussian random variables. These bounds are applied to Toeplitz and random matrices as well as to nonstationary AR(1) processes.

#### Article information

**Source**

Ann. Probab., Volume 27, Number 2 (1999), 1072-1098.

**Dates**

First available in Project Euclid: 29 May 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1022677395

**Digital Object Identifier**

doi:10.1214/aop/1022677395

**Mathematical Reviews number (MathSciNet)**

MR1699003

**Zentralblatt MATH identifier**

0941.60049

**Subjects**

Primary: 60F05: Central limit and other weak theorems

**Keywords**

Independent random variables quadratic forms asymptotic distribution limit theorems Berry–Esseen bounds

#### Citation

Götze, F.; Tikhomirov, A. N. Asymptotic Distribution of Quadratic Forms. Ann. Probab. 27 (1999), no. 2, 1072--1098. doi:10.1214/aop/1022677395. https://projecteuclid.org/euclid.aop/1022677395