The Annals of Probability

The Limit Behavior of Elementary Symmetric Polynomials of i.i.d. Random Variables When Their Order Tends to Infinity

Péter Major

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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.

Article information

Source
Ann. Probab., Volume 27, Number 4 (1999), 1980-2010.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022874824

Digital Object Identifier
doi:10.1214/aop/1022874824

Mathematical Reviews number (MathSciNet)
MR1742897

Zentralblatt MATH identifier
0963.60021

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords
Limit theorems U-statistics saddlepoint method

Citation

Major, Péter. The Limit Behavior of Elementary Symmetric Polynomials of i.i.d. Random Variables When Their Order Tends to Infinity. Ann. Probab. 27 (1999), no. 4, 1980--2010. doi:10.1214/aop/1022874824. https://projecteuclid.org/euclid.aop/1022874824


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References

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