Asymptotic Distribution of Quadratic Forms and Applications

F. Götze () and A. Tikhomirov ()
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F. Götze: University of Bielefeld
A. Tikhomirov: Syktyvkar State University and Mathematical Department of IMM of the Russian Academy of Sciences

Journal of Theoretical Probability, 2002, vol. 15, issue 2, 423-475

Abstract: Abstract We consider the quadratic formsQ $$\sum\limits_{\mathop {1 \leqslant j,k \leqslant N}\limits_{j \ne k} } {a_{jk} } $$ X j X k+ $$\sum\limits_{j = 1}^N {a_{jj} } $$ (X j 2 -E X j 2 )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 (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (1)

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DOI: 10.1023/A:1014867011101

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