Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On the asymptotic normality of self-normalized sums

Philip S. Griffina1 and David M. Masona2

a1 Department of Mathematics, Syracuse University, Syracuse, N.Y. 13244-1150, U.S.A.

a2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.

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.

(Received May 18 1990)

(Revised September 04 1990)