On Khintchine type inequalities for k-wise independent Rademacher random variables

We consider Khintchine type inequalities on the pth moments of vectors of Nk-wise independent Rademacher random variables. We show that an analogue of Khintchine's inequality holds, with a constant N1∕2−k∕2p, when k is even. We then show that this result is sharp for k=2; in particular, a version of Khintchine's inequality for sequences of pairwise Rademacher variables cannot hold with a constant independent of N. We also characterize the cases of equality and show that, although the vector achieving equality is not unique, it is unique (up to law) among the smaller class of exchangeable vectors of pairwise independent Rademacher random variables. As a fortunate consequence of our work, we obtain similar results for 3-wise independent vectors.