Probability Theory and Related Fields

, Volume 75, Issue 2, pp 261–277

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 Xiindependent (not necessarily identically distributed), and wijn(·, ·) Borel functions, such that wijn(Xi, Xj) 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 wijn(Xi, Xj) and an eigenvalue condition are given that ensure asymptotic normality of W(n).


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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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