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

TL;DR

This paper investigates Khintchine type inequalities for vectors of k-wise independent Rademacher variables, establishing sharp bounds and characterizing equality cases, with implications for understanding independence structures in probabilistic inequalities.

Contribution

It extends Khintchine inequalities to k-wise independent Rademacher variables, providing sharp bounds for even k and characterizing equality cases, including for 3-wise independence.

Findings

Khintchine inequality holds with constant N^{1/2 - k/2p} for even k

The inequality cannot be independent of N for k=2

Unique exchangeable vectors achieve equality for pairwise independence

Abstract

We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ $k$-wise independent Rademacher random variables. We show that an analogue of Khintchine's inequality holds, with a constant $N^{1/2-k/2p}$, 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 \emph{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 exchangable vectors of pairwise independent Rademacher random variables. As a fortunate consequence of our work, we obtain similar results for $3$-wise independent vectors.