Random deletion does not affect asymptotic normality or quadratic negligibility
A sample of random vectors \(X_i\) is considered. The main theorem gives six equivalent conditions for a deletion scheme, for which the number of deleted points is bounded in probability, as the sample size \(n\to\infty\). The sample sum of the ``trimmed'' sample is asymptotically normal (after norming and centering) if and only if the original sample sum is asymptotically normal (with the same norming and centering). The original sample must be drawn from a distribution in the domain of attraction of the multivariate normal distribution. Let be \(V_n=\sum^n_{i=1} X_iX_i^T\). The \(X_i\) are quadratic negligible if \(\max_nX^T_iV_n^{-1}X_i @>P>>0\), \(1\leq i\leq n\), \(n\to\infty\). The random deletion has no effect on the quadratic neglibility. The quadratic neglibility of the deleted sample belongs to the conditions of the main theorem.
scientific article; zbMATH DE number 1085313
scientific article