## Abstract

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 σ^{2} 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 ∈ 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_{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.

Original language | English |
---|---|

Article number | 17 |

Pages (from-to) | 1-29 |

Journal | Theory of Computing |

Volume | 16 |

Issue number | 17 |

DOIs | |

State | Published - 2020 |

## Keywords

- Concentration
- Hashing
- K-wise independence
- Pseudorandomness
- Symmetric polynomials

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computational Theory and Mathematics