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What portion of the sample makes a partial sum asymptotically stable or normal?
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  • Published: April 1986

What portion of the sample makes a partial sum asymptotically stable or normal?

  • Sándor Csörgő1,
  • Lajos Horváth1 &
  • David M. Mason2 

Probability Theory and Related Fields volume 72, pages 1–16 (1986)Cite this article

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Summary

Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<α≦2 be given. We show that if at each stage n a number k n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.

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Author information

Authors and Affiliations

  1. Bolyai Institute, Szeged University, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Sándor Csörgő & Lajos Horváth

  2. Department of Mathematical Science, University of Delaware, 19716, Newark, Delaware, USA

    David M. Mason

Authors
  1. Sándor Csörgő
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  2. Lajos Horváth
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  3. David M. Mason
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Additional information

Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant

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Csörgő, S., Horváth, L. & Mason, D.M. What portion of the sample makes a partial sum asymptotically stable or normal?. Probab. Th. Rel. Fields 72, 1–16 (1986). https://doi.org/10.1007/BF00343893

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  • Received: 20 April 1984

  • Issue Date: April 1986

  • DOI: https://doi.org/10.1007/BF00343893

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Keywords

  • Distribution Function
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Order Statistic
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