Convergence results for sequences of quadratic forms

Let (XI,)be a sequence of independent random variables, and let Q n = where for each N,(an:,k)is a doubly indexed sequence of weights. The convergence and the rate of convergence of the sequence of quadratic forms {Q n } are studied. These quadratic forms are linear sums of dependent variables; however, their convergence properties are similar to those of linear sums of independent variables provided the variables have finite rth absolute moments with 0 < r 2.while the rate of convergence has not been obtained for r < 2, it is shown to be different from that of linear sums.