On construction ofk-wise independent random variables

Abstract

A 0–1probability space is a probability space (Ω, 2^Ω,P), where the sample space Ω⊂-{0, 1}ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .

We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size$m(n,k) = \left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} n \\ {k - 1} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} n \\ {k - 2} \\ \end{array} } \right) + ... + \left( {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right)$ always exists. We prove that for somep ₁,p ₂, ...,p _n ∈ [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size Ω(n _k ). For a very large degree of independence —k=[αn], for α>1/2- and allp _i =1/2, we prove a lower bound on the size of$2^n \left( {1 - \frac{1}{{2\alpha }}} \right)$. We also give explicit constructions ofk-wise independent 0–1 probability spaces.