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Asymptotic normality and subsequential limits of trimmed sums. (English) Zbl 0688.60016
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

MSC:
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
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