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Asymptotic Distribution of Quadratic Forms and Applications

Authors:	F. Götze A. Tikhomirov

Affiliation:	(1) Fakultät für Mathematik, Universität Bielefeld, University of Bielefeld, 33501 Bielefeld 1, Germany;(2) Fakulty of Mathematics, Syktyvkar State University and Mathematical Department of IMM of the Russian Academy of Sciences, Oktjabrskyi prospekt 55, 167001 Syktyvkar, Russia

Abstract:	We consider the quadratic formsQ $sumlimits_{mathop {1 leqslant j,k leqslant N}limits_{j ne k} } {a_{jk} }$ X_jX_k+ $sumlimits_{j = 1}^N {a_{jj} }$ (X_j²-EX_j²)where X_j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a_jk) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.

Keywords:	independent random variables quadratic forms asymptotics of distribution limit theorems Berry– Esseen bounds
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