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≤ p≤∞, and Lorentz spaces Λψ. As a corollary, we obtain results that supplement the well-known Johnson--Schechtman theorem stating that the condition L_p ⊂ X, p < ∞ 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.