Inequalities and tail bounds for elementary symmetric polynomial with applications

We study the extent of independence needed to approximate the product of bounded random variables in expectation, a natural question that has applications in pseudorandomness and min-wise independent hashing. For random variables whose absolute value is bounded by $1$, we give an error bound of the form $\sigma^{\Omega(k)}$ where $k$ is the amount of independence and $\sigma^2$ is the total variance of the sum. Previously known bounds only applied in more restricted settings, and were quanitively weaker. We use this to give a simpler and more modular analysis of a construction of min-wise independent hash functions and pseudorandom generators for combinatorial rectangles due to Gopalan et al., which also slightly improves their seed-length. Our proof relies on a new analytic inequality for the elementary symmetric polynomials $Sk(x)$ for $x \in \mathbb{R}^n$ which we believe to be of independent interest. We show that if $|Sk(x)|,|S{k+1}(x)|$ are small relative to $|S{k-1}(x)|$ for some $k>0$ then $|S_\ell(x)|$ is also small for all $\ell > k$. From these, we derive tail bounds for the elementary symmetric polynomials when the inputs are only $k$-wise independent.