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- W3117043408 abstract "$ newcommand{R}{{mathbb R}} $ 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." @default.
- W3117043408 created "2021-01-05" @default.
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- W3117043408 date "2020-01-01" @default.
- W3117043408 modified "2023-10-09" @default.
- W3117043408 doi "https://doi.org/10.4086/toc.2020.v016a017" @default.
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