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.