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October 1999 The Limit Behavior of Elementary Symmetric Polynomials of i.i.d. Random Variables When Their Order Tends to Infinity
Péter Major
Ann. Probab. 27(4): 1980-2010 (October 1999). DOI: 10.1214/aop/1022874824

Abstract

Let $\xi_1,\xi_2\ldots$ be a sequence of i.i.d.random variables, and consider the elementary symmetric polynomial $S ^(k)(n)$ of order $k =k(n)$ of the first $n$ elements $\xi_1\ldots,\xi_n$ of this sequence. We are interested in the limit behavior of $S^(k) (n)$ with an appropriate transformation if $k(n)/n\rightarrow\alpha, 0<\alpha<1$. Since $k(n)\rightarrow\infty$ as $n\rightarrow\infty$, the classical methods cannot be applied in this case and new kinds of results appear.We solve the problem under some conditions which are satisfied in the generic case. The proof is based on the saddlepoint method and a limit theorem for sums of independent random vectors which mayhave some special interest in itself.

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Péter Major. "The Limit Behavior of Elementary Symmetric Polynomials of i.i.d. Random Variables When Their Order Tends to Infinity." Ann. Probab. 27 (4) 1980 - 2010, October 1999. https://doi.org/10.1214/aop/1022874824

Information

Published: October 1999
First available in Project Euclid: 31 May 2002

zbMATH: 0963.60021
MathSciNet: MR1742897
Digital Object Identifier: 10.1214/aop/1022874824

Subjects:
Primary: 60F05
Secondary: 60B15

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 4 • October 1999
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