Bounds on the sums of independent random variables in symmetric spaces (Q805041)

\textit{B. von Bahr} and \textit{C.-G. Esseen} [Ann. Math. Stat. 36, 299-303 (1965)] proved the inequality \[ E | \sum^{n}_{k=1}X_ k|^ p\leq (2-1/n)\sum^{n}_{k=1}E | X_ k|^ p \] for \(1\leq p\leq 2\), where \(X_ 1,X_ 2,...,X_ n\) are independent random variables with E \(X_ k=0\) and E \(| X_ k|^ p<\infty\). The author investigates this inequality for random variables from the symmetrical Banach spaces. He proves that this inequality (with the constant 2 instead of 2-1/n) is valid for pairwise disjoint random variables (i.e. \(X_ iX_ j=0\), \(i\neq j)\) if and only if it is valid for independent random variables in a wide class of symmetrical spaces.