Quadratic Negligibility and the Asymptotic Normality of Operator Normed Sums

✍ Scribed by R.A. Maller

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 711 KB
• Volume: 44
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1993.1011

✦ Synopsis

The condition (\max {1 \leqslant i \leqslant n} X{i}^{\top} V_{n}{ }^{1} X_{i} \xrightarrow{p} 0), where (X_{i}) are vectors in (R^{d}) and (V_{n}=\sum_{i=1}^{n} X_{i} X_{i}^{\mathrm{\top}}), is important in the asymptotics of various linear and nonlinear regression models. We call it "quadratic negligibility." It is shown that, when (X_{i}) are independent and identically distributed random vectors in (R^{d}), quadratic negligibility is equivalent to (X_{i}) 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 (V_{n}), along with results on the centering and norming constants for operator-normed convergence, are also given. (\quad 199.3) Academic Press. Inc.