Best constants in Rosenthal-type inequalities and the Kruglov operator

TL;DR

This paper establishes sharp constants in Rosenthal-type inequalities within symmetric Banach function spaces with the Kruglov property, linking these constants to the norm of the Kruglov operator, using a novel operator-based approach.

Contribution

It provides the first precise estimates of constants in Rosenthal inequalities in terms of the Kruglov operator norm, advancing the understanding of these inequalities in Banach spaces.

Findings

Sharp constants are equivalent to the Kruglov operator norm in the space.

New operator approach yields precise deterministic characterizations.

Results extend Rosenthal inequalities to a broader class of Banach spaces.

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 $X$.