Probability Theory and Related Fields

, Volume 75, Issue 2, pp 261–277

# A central limit theorem for generalized quadratic forms

• Peter de Jong
Article

## 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).

## Keywords

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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