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Elementary symmetric polynomials of increasing order
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  • Published: December 1988

Elementary symmetric polynomials of increasing order

  • A. J. van Es1 &
  • R. Helmers2 

Probability Theory and Related Fields volume 80, pages 21–35 (1988)Cite this article

  • 147 Accesses

  • 14 Citations

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Summary

The asymptotic behaviour of elementary symmetric polynomials S (k)n of order k, based on n independent and identically distributed random variables X 1,..., X n,is investigated for the case that both k and n get large. If \(k = o (n^{\frac{1}{2}} )\), then the distribution function of a suitably normalised S (k)n is shown to converge to a standard normal limit. The speed of this convergence to normality is of order \(\mathcal{O}(kn^{ - \frac{1}{2}} )\), provided \(k = \mathcal{O} (log^{ - 1} nlog_2^{ - 1} nn^{\frac{1}{2}} )\) and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k→∞ at the rate n 1/2 then a non-normal weak limit appears, provided the X i's are positive and S (k)n is standardised appropriately. On the other hand, if k→∞ at a rate faster than n 1/2 then it is shown that for positive X j'sthere exists no linear norming which causes S (k)n to converge weakly to a nondegenerate weak limit.

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Author information

Authors and Affiliations

  1. Mathematical Institute, University of Amsterdam, Roetersstraat 15, 1018 BW, Amsterdam, The Netherlands

    A. J. van Es

  2. Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB, Amsterdam, The Netherlands

    R. Helmers

Authors
  1. A. J. van Es
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  2. R. Helmers
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van Es, A.J., Helmers, R. Elementary symmetric polynomials of increasing order. Probab. Th. Rel. Fields 80, 21–35 (1988). https://doi.org/10.1007/BF00348750

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  • Received: 08 January 1986

  • Revised: 26 April 1988

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00348750

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Keywords

  • Distribution Function
  • Stochastic Process
  • Asymptotic Behaviour
  • Probability Theory
  • Normal Limit
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