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On the asymptotic normality of self-normalized sums

  • Philip S. Griffin (a1) and David M. Mason (a2)

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.

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[3] D. Darling . The influence of the maximal term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 (1952), 95107.

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[7] M. G. Hahn , J. Kuelbs and D. C. Weiner . The asymptotic distribution of magnitude-Winsorized sums via self-normalization. J. Theoretical Probability (1990), 137168.

[9] B. Logan , C. Mallows , S. Rice and L. Shepp . Limit distributions of self-normalized sums. Ann. Probab. 1 (1973), 788809.

[11] R.-D. Reiss . Uniform approximation to distributions of extreme order statistics. Adv. in Appl. Probab. 13 (1981), 533547).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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