The Annals of Probability
- Ann. Probab.
- Volume 27, Number 4 (1999), 1980-2010.
The Limit Behavior of Elementary Symmetric Polynomials of i.i.d. Random Variables When Their Order Tends to Infinity
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.
Ann. Probab., Volume 27, Number 4 (1999), 1980-2010.
First available in Project Euclid: 31 May 2002
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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