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Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

Author

Listed:
  • F. Götze

    (Universität Bielefeld)

  • A. N. Tikhomirov

    (Universität Bielefeld
    Faculty of Mathematics
    Syktyvkar State University)

Abstract

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 $${\cal {O}}(N^{-1})$$

Suggested Citation

  • F. Götze & A. N. Tikhomirov, 2005. "Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms," Journal of Theoretical Probability, Springer, vol. 18(4), pages 757-811, October.
    • Handle: RePEc:spr:jotpro:v:18:y:2005:i:4:d:10.1007_s10959-005-7525-3
    DOI: 10.1007/s10959-005-7525-3
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