Concentration for limited independence via inequalities for the elementary symmetric polynomials

We study the extent of independence needed to approximate the product of bounded random variables in expectation. This natural question has applications in pseudo-randomness and min-wise independent hashing. For random variables with absolute value bounded by 1, we give an error bound of the form σ^Ω(k) when the input is k-wise independent and σ² is the variance of their sum. Previously, known bounds only applied in more restricted settings, and were quantitatively weaker. Our proof relies on a new analytic inequality for the elementary symmetric polynomials S_k (x) for x ∈ Rⁿ. 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_ℓ (x)| is also small for all ℓ > 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.