Sums of Independent Random Variables in Rearrangement Invariant Function Spaces

Abstract

Let $X$ be a quasinormed rearrangement invariant function space on (0, 1) which contains $L_q(0, 1)$ for some finite $q$. There is an extension of $X$ to a quasinormed rearrangement invariant function space $Y$ on $(0, \infty)$ so that for any sequence $(f_i)^\infty_{i = 1}$ of symmetric random variables on (0,1), the quasinorm of $\sum f_i$ in $X$ is equivalent to the quasinorm of $\sum\mathbf{f}_i$ in $Y$, where $(\mathbf{f}_i)^\infty_{i = 1}$ is a sequence of disjoint functions on $(0, \infty)$ such that for each $i, \mathbf{f}_i$ has the same decreasing rearrangement as $f_i$. When specialized to the case $X = L_q(0, 1)$, this result gives new information on the quantitative local structure of $L_q$.