Concentration for limited independence via inequalities for the elementary symmetric polynomials

Parikshit Gopalan, Amir Yehudayoff

Research output: Contribution to journalArticlepeer-review


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 σΩ(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 Sk (x) for x ∈ Rn. We show that if |Sk (x)|, |Sk+1 (x)| are small relative to |Sk−1 (x)| for some k > 0 then |S (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 languageEnglish
Article number17
Pages (from-to)1-29
JournalTheory of Computing
Issue number17
StatePublished - 2020


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

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computational Theory and Mathematics


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