Asymptotic distribution of quadratic forms
The authors consider quadratic forms \(Q_n= \sum_{1\leq j\neq k\leq n}a_{jk} x_jx_k\), where \(x_j\) are i.i.d. random variables. They obtain optimal bounds for the Kolmogorov distance between the distribution of \(Q_n\) and the distribution \(G_n\) of the same quadratic forms with \(x_j\) replaced by corresponding orthonormal Gaussian random variables \(y_j\). Early, \textit{V. I. Rotar'} [Theory Probab. Appl. 18, 499-507 (1973); translation from Teor. Veroyatn. Primen. 18, 527-534 (1973; Zbl 0304.60037)] proved that under sufficiently weak conditions on the matrix \(A= \{a_{jk}\}^n_{j,k=1}\) and for large \(n\), the distribution of \(Q_n\) is close to the distribution of \(G_n\). \textit{N. G. Gamkrelidze} and \textit{V. I. Rotar'} [ibid. 22, 394-397 (1977), resp. ibid. 22, 404-407 (1977; Zbl 0385.60035)] and \textit{V. I. Rotar'} and \textit{T. L. Shervashidze} [ibid. 30, 585-590 (1986), resp. ibid. 30, No. 3, 549-554 (1985; Zbl 0578.60020)] obtained bounds for the error of this approximation. The aim of this paper is to improve this bound for i.i.d. r.v.'s \(x_1,x_2,\dots, x_n\) under various assumptions on the spectra of the matrices \(A\). Moreover, these bounds are applied to Toeplitz and random matrices as well as to nonstationary \(\text{AR}(1)\) process.
scientific article; zbMATH DE number 1359657