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On the asymptotic normality of self-normalized sums

Published online by Cambridge University Press:  24 October 2008

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.

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
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Bingham, N., Teugels, J. and Goldie, C.. Regular Variation (Cambridge University Press, 1987).Google Scholar
[2]Csörgő, S., Haeusler, E. and Mason, D. M.. The quantile-transform approach to the asymptotic distribution of modulus trimmed sums. Sums, Trimmed Sums and Extremes (eds. Holm, M. G., Mason, D. M. and Weiner, D. C.). Birkhäuser, Boston (1990).Google Scholar
[3]Darling, D.. The influence of the maximal term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 (1952), 95107.CrossRefGoogle Scholar
[4]Efron, B.. Student's t-test under symmetry conditions. J. Amer. Statist. Assoc. (1969) 12781302.Google Scholar
[5]Gnedenko, B. V. and Kolmogorov, A. N.. Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, 1954).Google Scholar
[6]Griffin, P. S. and Pruitt, W. E.. The central limit problem for trimmed sums. Math. Proc. Cambridge Philos. Soc. 102 (1987), 329349.CrossRefGoogle Scholar
[7]Hahn, M. G., Kuelbs, J. and Weiner, D. C.. The asymptotic distribution of magnitude-Winsorized sums via self-normalization. J. Theoretical Probability (1990), 137168.CrossRefGoogle Scholar
[8]Hahn, M. G. and Weiner, D. C.. Asymptotic behavior of self-normalized trimmed sums: nonnormal limits. (Preprint, 1989.)Google Scholar
[9]Logan, B., Mallows, C., Rice, S. and Shepp, L.. Limit distributions of self-normalized sums. Ann. Probab. 1 (1973), 788809.Google Scholar
[10]Mori, T.. On the limit distribution of lightly trimmed sums. Math. Proc. Cambridge Philos. Soc. 96 (1984), 507516.CrossRefGoogle Scholar
[11]Reiss, R.-D.. Uniform approximation to distributions of extreme order statistics. Adv. in Appl. Probab. 13 (1981), 533547).CrossRefGoogle Scholar