Necessary and Sufficient Conditions for Asymptotic Normality of $L$-Statistics

Abstract

It is now classical that the sample mean $\bar{Y}$ is known to be asymptotically normal with $\sqrt n$ norming if and only if $0 < \operatorname{Var}\lbrack Y\rbrack < \infty$ and with arbitrary norming if and only if the df of $Y$ is in the domain of attraction of the normal df. Now let $T_n = n^{-1}\sum c_{ni}h(X_{n:i})$ for order statistics $X_{n:i}$ from a $\operatorname{df} F$ denote a general $L$-statistic subject to a bit of regularity; the key condition introduced into this problem in this paper is the regular variation of the score function $J$ defining the $c_{ni}$'s. We now define a rv $Y$ by $Y = K(\xi)$, where $\xi$ is uniform (0, 1) and where $dK = J dh(F^{-1})$. Then $T_n$ is shown to be asymptotically normal with $\sqrt n$ norming if and only if $0 < \operatorname{Var}\lbrack Y\rbrack < \infty$ and with arbitrary norming if and only if the df of $Y$ is in the domain of attraction of the normal df. As it completely parallels the classical theorem, this theorem gives the right conclusion for $L$-statistics. In order to establish the necessity above, we also obtain a nice necessary and sufficient condition for the stochastic compactness of $T_n$ and give a representation formula for all possible subsequential limit laws.