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Asymptotic Distribution of Quadratic Forms and Applications

Author

Listed:
  • F. Götze

    (University of Bielefeld)

  • A. Tikhomirov

    (Syktyvkar State University and Mathematical Department of IMM of the Russian Academy of Sciences)

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.

Suggested Citation

  • F. Götze & A. Tikhomirov, 2002. "Asymptotic Distribution of Quadratic Forms and Applications," Journal of Theoretical Probability, Springer, vol. 15(2), pages 423-475, April.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:2:d:10.1023_a:1014867011101
    DOI: 10.1023/A:1014867011101
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    Cited by:

    1. Ángel Felipe & María Jaenada & Pedro Miranda & Leandro Pardo, 2023. "Restricted Distance-Type Gaussian Estimators Based on Density Power Divergence and Their Applications in Hypothesis Testing," Mathematics, MDPI, vol. 11(6), pages 1-41, March.

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