Journal of Theoretical Probability

, Volume 18, Issue 4, pp 757–811 | Cite as

Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

  • F. Götze
  • A. N. Tikhomirov
We consider quadratic forms of the type
$$ Q(F,{\bf A})=\sum_{\mathop{1\le j,k \le N}\limits_{j\ne k}}a_{jk} X_j X_k, $$
where X j are i.i.d. random variables with common distribution F and finite fourth moment, \({\bf A}=\{a_{jk}\}_{j,k=1}^N\) denotes a symmetric matrix with eigenvalues λ1, ..., λ N 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 \({\cal {O}}(N^{-1})\)

Key words

Non-central limit theorems quadratic forms random vectors 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeld 1Germany
  2. 2.Faculty of MathematicsSyktyvkarRussia
  3. 3.The Mathematical Department of IMM of the Ural Branch of Russian Academia of SciencesSyktyvkar State UniversityRussia

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