On construction ofk-wise independent random variables
AbstractA 0–1probability space is a probability space (Ω, 2Ω,P), where the sample space Ω⊂-{0, 1}n for somen. A probability space isk-wise independent if, whenYi is defined to be theith coordinate or the randomn-vector, then any subset ofk of theYi's is (mutually) independent, and it is said to be a probability spacefor p1,p2, ...,pn ifP[Yi=1]=pi.We study constructions ofk-wise independent 0–1 probability spaces in which thepi's are arbitrary. It was known that for anyp1,p2, ...,pn, 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 somep1,p2, ...,pn ∈ [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 eachpi=k/n has size Ω(nk). For a very large degree of independence —k=[αn], for α>1/2- and allpi=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.