## 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

#### 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

#### 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

#### References

• [1] Dynkin, E. B. and Mandelbaum, A. (1983). Symmetric statistics, Poisson processes and multiple Wiener integrals. Ann.Statist.11 739-745.
• [2] Hal´asz, G. and Sz´ekely, G. J. (1976). On the elementary symmetric polynomials of independent random variables. Acta Math.Acad.Sci.Hungar.28 397-400.
• [3] M ´ori, T. F. and Sz´ekely, G. J. (1982). Asymptotical behavior of symmetric polynomial statistics. Ann.Probab.10 124-131.
• [4] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York.
• [5] Raugi, A. (1979). Th´eor eme de la limite centrale pour un produit semi-direct d'un groupe de Lie r´esoluble simplement connexe de type rigide par un groupe compact. Probability Measures on Groups. Lecture Notes in Math. 706 257-324. Springer, Berlin.
• [6] van Ess, G. (1986). On the weak limit of elementary symmetric polynomials. Ann.Probab. 14 667-695.