Let $\Vert\cdot\Vert$ is a norm in $R^n$. Let $x_1,\dots,x_N$ are non-necessarily independent Rademacher random variables random variables (variables which are uniform on $\{-1, 1\}$). By $E$ we denote an expectation. There exists a constant $A_{p,q}$ depending only on $p,q$ such that for any vectors $u_1,\dots, u_N\in R^n$ $$ \left(E \left\Vert\sum_{i=1}^N x_iu_i\right\Vert^p\right)^\frac{1}{p}\leq A_{p,q} E \left(\left\Vert\sum_{i=1}^N x_iu_i\right\Vert^{q}\right)^{\frac 1q}, $$ for all $p,q \in [1, \infty)$.
This is so called restricted Kahane's inequality.
What are the possible applications of this inequality?