Geodesic
Comparison of Sums of Independent and Disjoint Functions in Symmetric Spaces
Matematičeskie zametki, Tome 76 (2004) no. 4, pp. 483-489.

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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. 
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S. V. Astashkin; F. A. Sukochev. Comparison of Sums of Independent and Disjoint Functions in Symmetric Spaces. Matematičeskie zametki, Tome 76 (2004) no. 4, pp. 483-489. http://geodesic.mathdoc.fr/item/MZM_2004_76_4_a0/

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