On construction of k-wise independent random variables

Howard Karloff, Yishay Mansour

Research output: Contribution to journalArticlepeer-review

Abstract

A 0-1 probability space is a probability space (Ω,2Ω,P), where the sample space Ω ⊆ {0, 1}n for some n. A probability space is k-wise independent if, when Yi is defined to be the ith coordinate of the random n-vector, then any subset of k of the Yi's is (mutually) independent, and it is said to be a probability space for p1,p2, . . ., pn if P[Yi = 1] = pi. We study constructions of k-wise independent 0-1 probability spaces in which the pi's are arbitrary. It was known that for any p1, p2, . . . , pn, a k-wise independent probability space of size (matrix presented) always exists. We prove that for some p1, p2, . . ., pn ∈ [0,1], m(n,k) is a lower bound on the size of any k-wise independent 0-1 probability space. For each fixed k, we prove that every k-wise independent 0-1 probability space when each pi-k/n has size Ω(nk). For a very large degree of independence - k = ⌊αn⌋, for α > 1/2 - and all pi, = 1/2, we prove a lower bound on the size of 2n(1-1/2α). We also give explicit constructions of k-wise independent 0-1 probability spaces.

Original languageEnglish
Pages (from-to)91-107
Number of pages17
JournalCombinatorica
Volume17
Issue number1
DOIs
StatePublished - 1997

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