TY - JOUR

T1 - On construction of k-wise independent random variables

AU - Karloff, Howard

AU - Mansour, Yishay

N1 - Funding Information:
Mathematics Subject Classification (1991): 68 Q 99, 68 R 05, 60 C 05 * This author was supported in part by NSF grant CCR 9107349. ** This research was supported in part by the Israel Science Foundation administered by the Israel Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.

PY - 1997

Y1 - 1997

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0031459086&partnerID=8YFLogxK

U2 - 10.1007/BF01196134

DO - 10.1007/BF01196134

M3 - מאמר

AN - SCOPUS:0031459086

VL - 17

SP - 91

EP - 107

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -