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On the asymptotic normality of self-normalized sums. (English) Zbl 0723.62008

Summary: Let \(X_ 1,...,X_ n\) be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let \(S_ n(r_ n)\) denote their sum when the \(r_ n\) largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the Studentized version of \(S_ n(r_ n)\), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when \(r_ n=r\) these conditions are the same, but when \(r_ n\to \infty\), the former holds more generally.

MSC:

62E20 Asymptotic distribution theory in statistics
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