On K-wise Independent Distributions and Boolean Functions

Abstract: We pursue a systematic study of the following problem. Let f:{0,1}^n -> {0,1} be a (usually monotone) Boolean function whose behaviour is well understood when the input bits are identically independently distributed. What can be said about the behaviour of the function when the input bits are not completely independent, but only k-wise independent, i.e. every subset of k bits is independent? more precisely, how high should k be so that any k-wise independent distribution "fools" the function, i.e. causes it to behave nearly the same as when the bits are completely independent? We analyze several well known Boolean functions (including AND, Majority, Tribes and Percolation among others), some of which turn out to have surprising properties. In some of our results we use tools from the theory of the classical moment problem, seemingly for the first time in this subject, to shed light on these questions.