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August, 1981 Convergence to a Stable Distribution Via Order Statistics
Raoul LePage, Michael Woodroofe, Joel Zinn
Ann. Probab. 9(4): 624-632 (August, 1981). DOI: 10.1214/aop/1176994367


Let $X_1, X_2, \cdots$ be i.i.d. random variables whose common distribution function $F$ is in the domain of attraction of a nonnormal stable distribution. A simple, probabilistic proof of the convergence of the normalized partial sums to the stable distribution is given. The proof makes use of an elementary property of order statistics and clarifies the manner in which the largest few summands determine the limiting distribution. The method is applied to determine the limiting distribution of self-norming sums and deduce a representation for the limiting distribution. The representation affords an explanation of the infinite discontinuities of the limiting densities which occur in some cases. Application of the technique to prove weak convergence in a separable Hilbert space is explored.


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Raoul LePage. Michael Woodroofe. Joel Zinn. "Convergence to a Stable Distribution Via Order Statistics." Ann. Probab. 9 (4) 624 - 632, August, 1981.


Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0465.60031
MathSciNet: MR624688
Digital Object Identifier: 10.1214/aop/1176994367

Primary: 60F05
Secondary: 60B99

Keywords: norming sums , order statistics , random signs , separable Banach and Hilbert Spaces , Stable distributions

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 4 • August, 1981
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