Best constants in Rosenthal-type inequalities and the Kruglov operator

Abstract

Let $X$ be a symmetric Banach function space on $[0,1]$ with the Kruglov property, and let $\mathbf{f}=\{f_k\}_{{k=1}}^n$, $n\ge1$ be an arbitrary sequence of independent random variables in $X$. This paper presents sharp estimates in the deterministic characterization of the quantities \[\Biggl\|\sum_{{k=1}}^nf_k\Biggr\|_X,\Biggl\|\Biggl(\sum_{{k=1}}^n|f_k|^p\Biggr)^{1/p}\Biggr\|_X,\qquad 1\leq p<\infty,\] in terms of the sum of disjoint copies of individual terms of $\mathbf{f}$. Our method is novel and based on the important recent advances in the study of the Kruglov property through an operator approach made earlier by the authors. In particular, we discover that the sharp constants in the characterization above are equivalent to the norm of the Kruglov operator in $X$.