# On construction of k-wise independent random variables howard

Howard Karloff, Yishay Mansour

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

A O-1 probability space is a probability space (Ω, 2Ω, P), where the sample space Ω ⊆ {O, 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 U's is (mutually) independent. We say it is a probability space for PI, p2, . . . . pn if P[Yi = 1] = pi. We study constructions of k-wise independent O-1 probability spaces in which the pi's are arbitrary. It was known that for any P1, pZ, . . . . pn, a k-wise independent probability space of size m(n, k) = (n/k) + (k-l) + (nk/2) +...+n/0 always exists. We prove that for some P1,P2) . ..1 Pn ϵ [0, 1], Ω(n, k) is a lower bound on the size of any k-wise independent O-1 probability space. For each fixed k, for each n we prove the existence of a p c [0, 1], depending on n, such that every k-wise independent O-1 probability space if all pi = p has size fl(n). (This is in contrast with the known construction of size O(n(k/2) when all the probabilities are 1/2.) For a very large degree of independence = [an], for cs > l/2-And all pi = 1/2, we prove a lower bound on the size of 2n (1 -1/2α), in contrast with the previous lower bound of βn where β < 2 is a function only of a. We give explicit constructions of k-wise independent O-1 probability spaces with arbitrary pi, and tight constructions for pairwise spaces with identical pi's.

Original language English Proceedings of the 26th Annual ACM Symposium on Theory of Computing, STOC 1994 Association for Computing Machinery 564-573 10 0897916638 https://doi.org/10.1145/195058.195409 Published - 23 May 1994 26th Annual ACM Symposium on Theory of Computing, STOC 1994 - Montreal, CanadaDuration: 23 May 1994 → 25 May 1994

### Publication series

Name Proceedings of the Annual ACM Symposium on Theory of Computing Part F129502 0737-8017

### Conference

Conference 26th Annual ACM Symposium on Theory of Computing, STOC 1994 Canada Montreal 23/05/94 → 25/05/94

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