Revised: August 30, 2020

Published: December 24, 2020

**Keywords:**pseudorandomness, $k$-wise independence, hashing, concentration, symmetric polynomials

**Categories:**pseudorandomness, $k$-wise independence, hashing, concentration, polynomials, polynomials - multivariate, inequalities, inequalities for elementary symmetric polynomials

**ACM Classification:**G.3

**AMS Classification:**68Q87

**Abstract:**
[Plain Text Version]

We study the extent of independence needed to approximate the product of bounded random variables in expectation. This natural question has applications in pseudorandomness and min-wise independent hashing.

For random variables with absolute value bounded by $1$, we give an error bound of the form $\sigma^{\Omega(k)}$ when the input is $k$-wise independent and $\sigma^2$ is the variance of their sum. Previously, known bounds only applied in more restricted settings, and were quantitively weaker. Our proof relies on a new analytic inequality for the elementary symmetric polynomials $S_k(x)$ for $x \in \R^n$. We show that if $|S_k(x)|,|S_{k+1}(x)|$ are small relative to $|S_{k-1}(x)|$ for some $k> 0$ then $|S_\ell(x)|$ is also small for all $\ell > k$. 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 improves the overall seed-length.