## Abstract

A 0–1*probability space* is a probability space (Ω, 2^{Ω},*P*), where the sample space Ω⊂-{0, 1}^{n} for some*n*. A probability space is*k-wise independent* if, when*Y*
_{
i
} is defined to be the*i*th coordinate or the random*n*-vector, then any subset of*k* of the*Y*
_{
i
}'s is (mutually) independent, and it is said to be a probability space*for p*
_{1},*p*
_{2}, ...,*p*
_{
n
} if*P*[*Y*
_{
i
}=1]=*p*
_{
i
}.

We study constructions of*k*-wise independent 0–1 probability spaces in which the*p*
_{
i
}'s are arbitrary. It was known that for any*p*
_{1},*p*
_{2}, ...,*p*
_{
n
}, a*k*-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 some*p*
_{1},*p*
_{2}, ...,*p*
_{
n
} ∈ [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*p*
_{
i
}=*k/n* has size Ω(*n*
_{
k
}). For a very large degree of independence —*k*=[α*n*], for α>1/2- and all*p*
_{
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 of*k*-wise independent 0–1 probability spaces.

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### Affiliations

## Additional information

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 lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.

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### Cite this article

Karloff, H., Mansour, Y. On construction of*k*-wise independent random variables.
*Combinatorica* **17, **91–107 (1997). https://doi.org/10.1007/BF01196134

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### Mathematics Subject Classification (1991)

- 68 Q 99
- 68 R 05
- 60 C 05