# Probabilistic results on the elementary symmetric polynomials

The elementary symmetric polynomials of degree $k$ in $N$ variables are defined as

$$e_k(x_1, \ldots, x_N) = \sum_{(i_1,\ldots,i_N) \in I_k^N}{x_1^{i_1}\ldots x_N^{i_N}}, \quad 0 \le k \le N$$

with $I_k^N = \{(i_1, \ldots, i_N): i_n \in \{0, 1\},\ \sum_{n = 0}^N i_n = k\}$. For a given $k$, this is basically the sum of all different products of $k$ of the $N$ variables. The elementary symmetric polynomials are also related to the coefficients of a univariate polynomial with roots $x_1, \ldots, x_N$ in the following way

$$\prod_{n = 1}^{N}(t - x_n) = \sum_{k = 0}^{N}{(-1)^ke_k(x_1, \ldots, x_N)t^{N - k}}$$

What I'm wondering is if there are any bounds and/or probabilistic results about the polynomials $e_k$. That is, if you take $x_1, \ldots, x_N \in \mathbb{R}$ to be independent and identically distributed according to some distribution, (a zero mean Gaussian with some variance, for example) can you find the PDF of $e_k$, or less ambitiously, give a second order characterization of $e_k$? Or if you know the $x_1, \ldots, x_N$, can you bound $e_k?$ And if $x_1, \ldots, x_N$ are complex instead of real?

It's not much of a concrete question. if anyone knows of any results with this flavour, I would be happy to know.

Thanks.

-
Could you make the question more precise? –  Did Feb 16 '13 at 10:05