Griffin, Philip S.; Pruitt, William E. Asymptotic normality and subsequential limits of trimmed sums. (English) Zbl 0688.60016 Ann. Probab. 17, No. 3, 1186-1219 (1989). Let \((X_ n)_{n\geq 1}\) be a sequence of nondegenerate i.i.d. random variables, and for each \(n\geq 1\) let \(X_{n1}\leq X_{n2}\leq...\leq X_{nn}\) denote the order statistics of \(X_ 1,X_ 2,...,X_ n\). Moreover, let \((r_ n)_{n\geq 1}\) and \((s_ n)_{n\geq 1}\) be two sequences of nonnegative integers such that \(r_ n\to \infty\), \(s_ n\to \infty\), \(r_ n/n\to 0\) and \(s_ n/n\to 0\) as \(n\to \infty\). The authors consider the trimmed sums \[ S_ n(s_ n,r_ n)=\sum^{n-r_ n}_{k=s_ n+1}X_{nk}, \] which are the usual partial sums \(S_ n=\sum^{n}_{k=1}X_ k\) with the \(s_ n\) smallest and \(r_ n\) largest summands discarded. They obtain a necessary and sufficient condition for asymptotic normality of \(S_ n(s_ n,r_ n)\) and a complete description of the class of all possible subsequential limit laws for \(S_ n(s_ n,r_ n)\). They also give sufficient conditions for convergence of a particular subsequence to a given limit which will be necessary as well if a result on unique factorization of characteristic functions can be proved. The methods of proof rely on the theorems of Berry-Esseen and Lyapunov. Using a quantile function - empirical process methodology the problem of the asymptotic distribution of the trimmed sums \(S_ n(s_ n,r_ n)\) was also studied by S. Csörgö, the reviewer and D. M. Mason [ibid. 16, No.2, 672-699 (1988; Zbl 0647.62030)]. Reviewer: E.Häusler Cited in 19 Documents MSC: 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics 62G30 Order statistics; empirical distribution functions Keywords:stochastic compactness; discarding outliers; order statistics; trimmed sums; asymptotic normality; subsequential limit laws; factorization of characteristic functions; Berry-Esseen; Lyapunov PDF BibTeX XML Cite \textit{P. S. Griffin} and \textit{W. E. Pruitt}, Ann. Probab. 17, No. 3, 1186--1219 (1989; Zbl 0688.60016) Full Text: DOI