On the relationship between ε-biased random variables and ε-dependent random variables

The notions of "k-wise ε-dependent" and "k-wise ε-biased" are somewhat weaker in randomness than those of independent random variables. Random variables with these properties could be substitutes for independent random variables of randomized algorithms. In this paper, after giving relevant definitions for these notions, the relationship between these notions is presented: For any integer k and n with 1 ≤ k ≤ n, if a system of n random variables is k-wise ε-biased, then it is k-wise 4(1 - 2^-k)ε-dependent in maximum norm and k-wise 2(1 - 2^-k)ε-dependent in L₁ norm with respect to the uniform distribution. It has been presented, in literature, that k-wise ε-biased random variables are substituted for k-wise δ-dependent random variables in many randomized algorithms, so the results of this paper are expected to reduce the running time of resultant algorithms due to derandomization.