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On the asymptotic normality of self-normalized sums

Published online by Cambridge University Press:  24 October 2008

Philip S. Griffin  and
David M. Mason
Philip S. Griffin
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, N.Y. 13244-1150, U.S.A.
David M. Mason
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.
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Abstract

Let X1, …, Xn be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let Sn(rn) denote their sum when the rn largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the studentized version of Sn(rn), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when rn = r these conditions are the same, but when rn → ∞the former holds more generally.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 3 , May 1991 , pp. 597 - 610
Copyright
Copyright © Cambridge Philosophical Society 1991

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