On Khinchine type inequalities for pairwise independent Rademacher random variables

TL;DR

This paper investigates Khintchine type inequalities for pairwise independent Rademacher variables, showing the best possible constants depend on the number of variables and establishing sharp bounds for exchangeable cases.

Contribution

It proves that Khintchine inequalities cannot have N-independent constants in this setting and determines the optimal constant growth rate, including sharp bounds for exchangeable vectors.

Findings

Best constant is at least N^{1/2-1/p}

Sharp bounds established for exchangeable vectors when p=4

Results extend to 3-wise independent vectors

Abstract

We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ pairwise independent Rademacher random variables. We establish that an analogue of Khintchine's inequality cannot hold in this setting with a constant that is independent of $N$; in fact, we prove that the best constant one can hope for is at least $N^{1/2-1/p}$. Furthermore, we show that this estimate is sharp for exchangeable vectors when $p = 4$. As a fortunate consequence of our work, we obtain similar results for $3$-wise independent vectors.