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.
Citation
F. Götze. A. N. Tikhomirov. "Asymptotic Distribution of Quadratic Forms." Ann. Probab. 27 (2) 1072 - 1098, April 1999. https://doi.org/10.1214/aop/1022677395
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