TY - JOUR

T1 - On the asymptotic normality of self-normalized sums

AU - Griffin, Philip S.

AU - Mason, David M.

N1 - Funding Information:
The first author's research is supported in part by NSF Grant DMS-8700928 and the second author's research is supported in part by NSF Grant DMS-8803209.

PY - 1991

Y1 - 1991

N2 - Let Xj, …,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.

AB - Let Xj, …,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.

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U2 - 10.1017/S0305004100070018

DO - 10.1017/S0305004100070018

M3 - Article

AN - SCOPUS:84974306811

VL - 109

SP - 597

EP - 610

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -