The Annals of Probability

Asymptotic Distribution of Quadratic Forms

F. Götze and A. N. Tikhomirov

Full-text: Open access

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


Export citation