Orlicz norms of sequences of random variables

Abstract

Let $f_{i}$, $i=1,\dots,n$, be copies of a random variable f and let N be an Orlicz function. We show that for every $x\in \mathbb{R}^{n}$ the expectation $\mathbf{E} \| (x_i f_i) _{i=1}^n \|_N $ is maximal (up to an absolute constant) if $f _{i}$, $i=1,\dots,n$, are independent. In that case we show that the expectation $\mathbf{E} \| (x_i f_i) _{i=1}^n \| _N $ is equivalent to $\|x\| _M$, for some Orlicz function M depending on N and on distribution of f only. We provide applications of this result.