Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms |
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| Authors: | F.?G?tze mailto:goetze@math.uni-bielefeld.de" title=" goetze@math.uni-bielefeld.de" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author,A.?N.?Tikhomirov |
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| Affiliation: | 1.Fakult?t für Mathematik,Universit?t Bielefeld,Bielefeld 1,Germany;2.Faculty of Mathematics,Syktyvkar,Russia;3.The Mathematical Department of IMM of the Ural Branch of Russian Academia of Sciences,Syktyvkar State University,Russia |
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| Abstract: | ![]()
We consider quadratic forms of the type where Xj are i.i.d. random variables with common distribution F and finite fourth moment,  denotes a symmetric matrix with eigenvalues λ1, ..., λN ordered to be non-increasing in absolute value. We prove explicit bounds in terms of sums of 4th powers of entries of the matrix A and the size of the eigenvalue λ13 for the approximation of the distribution of Q(F,A) by the distribution of Q (φ, A) where φ is standard Gaussian distribution. In typical cases this error is of optimal order  Supported by the DFG-Forschergruppe FOR 399/1-1 at Bielefeld. Partially supported by INTAS N 03-51-5018. Partially supported by RFBR and RFBR–DFG, grants NN 02-01-00233, 04-01-04000 |
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| Keywords: | Non-central limit theorems quadratic forms random vectors |
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