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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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