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Asymptotic expansion in the central limit theorem for quadratic forms

Authors:

F. Götze A. N. Tikhomirov V. A. Yurchenko

Affiliation:

1.Fakult?t für Mathematik,Universit?t Bielefeld,Germany;2.Faculty of Mathematics,Syktyvkar University,Syktyvkar,Russia

Abstract:

We consider statistics of the form

$Q_n = sumlimits_{j = 1}^N {a_{jj} } (X_j^2 - mu _2 ) + sumlimits_{1 leqslant j ne k leqslant N} {a_{jk} } X_j X_k$

, where the X_j are i.i.d. random variables with finite sixth moment. We obtain the rate of convergence in the central limit theorem for the one-term Edgeworth expansion. Furthermore, applications to Toeplitz matrices, quadratic form of ARMA-processes, goodness-of-fit, as well as spacing statistics are included. Bibliography: 16 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 81–114.

Keywords:

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