Symmetric quasi-norms of sums of independent random variables in symmetric function spaces with the Kruglov property

Abstract

Let X be a symmetric Banach function space on [0, 1] and let E be a symmetric (quasi)-Banach sequence space. Let f = {fk}k=1n, n ≥ 1 be an arbitrary sequence of independent random variables in X and let {ek}k=1∞ ⊂ E be the standard unit vector sequence in E. This paper presents a deterministic characterization of the quantity $$||||\sum\limits_{k = 1}^n {{f_k}{e_k}|{|_E}|{|_X}} $$ in terms of the sum of disjoint copies of individual terms of f. We acknowledge key contributions by previous authors in detail in the introduction, however our approach is based on the important recent advances in the study of the Kruglov property of symmetric spaces made earlier by the authors. Authors acknowledge support from the ARC.