Quadratic Negligibility and the Asymptotic Normality of Operator Normed Sums

The condition max1≤i≤nXTiV−1nXi[formula] 0, where Xi are vectors in Rd and Vn = ∑ni=1XiXTi, is important in the asymptotics of various linear and nonlinear regression models. We call it "quadratic negligibility." It is shown that, when Xi are independent and identically distributed random vectors in Rd, quadratic negligibility is equivalent to Xi being in the operator normed domain of attraction of the multivariate normal distribution, thereby generalising the one-dimensional case. Related results on the convergence of the matrix Vn, along with results on the centering and norming constants for operator-normed convergence, are also given.