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A central limit theorem for generalized quadratic forms
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  • Published: 01 June 1987

A central limit theorem for generalized quadratic forms

  • Peter de Jong1 

Probability Theory and Related Fields volume 75, pages 261–277 (1987)Cite this article

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Summary

Random variables of the form \(W(n) = \mathop \sum \limits_{1 \leqq i \leqq n} \mathop \sum \limits_{{\text{ }}1 \leqq j \leqq n} w_{ijn} (X_i ,X_j )\) are considered with X i independent (not necessarily identically distributed), and w ijn (·, ·) Borel functions, such that w ijn (X i , X j ) is square integrable and has vanishing conditional expectations:

$$E(w_{ijn} (X_i ,X_j )|X_i = E(w_{ijn} (X_i ,X_j )|X_j ) = 0,{\text{ a}}{\text{.s}}{\text{.}}$$

A central limit theorem is proved under the condition that the normed fourth moment tends to 3. Under some restrictions the condition is also necessary. Finally conditions on the individual tails of w ijn (X i , X j ) and an eigenvalue condition are given that ensure asymptotic normality of W(n).

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Authors and Affiliations

  1. Mathematical Institute of the University of Amsterdam, Roetersstraat 15, 1018 WB, Amsterdam, The Netherlands

    Peter de Jong

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  1. Peter de Jong
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de Jong, P. A central limit theorem for generalized quadratic forms. Probab. Th. Rel. Fields 75, 261–277 (1987). https://doi.org/10.1007/BF00354037

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  • Received: 07 February 1986

  • Revised: 20 December 1986

  • Published: 01 June 1987

  • Issue Date: June 1987

  • DOI: https://doi.org/10.1007/BF00354037

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Keywords

  • Stochastic Process
  • Probability Theory
  • Quadratic Form
  • Limit Theorem
  • Statistical Theory
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