A central limit theorem for generalized quadratic forms

Abstract

Random variables of the form {Mathematical expression} 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: {Mathematical expression} 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). © 1987 Springer-Verlag.