Comparison of Sums of Independent and Disjoint Functions in Symmetric Spaces

The sums of independent functions (random variables) in a symmetric space $X$ on $[0,1]$ are studied. We use the operator approach closely connected with the methods developed, primarily, by Braverman. Our main results concern the Orlicz exponential spaces $\exp(L_p)$, $1\leqslant p\leqslant\infty$, and Lorentz spaces $\Lambda_\psi$. As a corollary, we obtain results that supplement the well-known Johnson–Schechtman theorem stating that the condition $L_p\subset X$, $p<\infty$, implies the equivalence of the norms of sums of independent functions and their disjoint “copies”. In addition, a statement converse, in a certain sense, to this theorem is proved.