Probability Theory and Related Fields

, Volume 75, Issue 2, pp 261–277 | Cite as

A central limit theorem for generalized quadratic forms

  • Peter de Jong


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).


Stochastic Process Probability Theory Quadratic Form Limit Theorem Statistical Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Peter de Jong
    • 1
  1. 1.Mathematical Institute of the University of AmsterdamAmsterdamThe Netherlands

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