On Rosenthal's inequality and rearrangement invariant spaces
This paper may be considered as a continuation of the author's previous paper [ibid. 32, No. 3, 376-382 (1991); resp. ibid. 32, No. 3(187), 31-38 (1991; Zbl 0756.46013)]. The author indicates a number of conditions under which independent random variables \(X_1, X_2, \dots, X_n\) from a rearrangement invariant space \({\mathcal L}\) of random variables satisfy the following inequalities \[ \left |\sum^n_{k = 1} X_k \right |_{\mathcal L} \leq C \max \left \{\left( \sum^n_{k = 1} |X_k |^p_{\mathcal L} \right)^{1/p}, \;\left( \sum^n_{k = 1} EX^2_k \right)^{1/2} \right\}, \] where the constant \(C\) depends only on \({\mathcal L}\), and \[ \left |\sum^n_{k = 1} X_k \right |_{\mathcal L} \leq B \left( \sum^n_{k = 1} |X_k |^p_{\mathcal L} \right)^{1/p}, \] where the constant \(B\) depends only on \({\mathcal L}\). In the both inequalities \(p\) is a number from \([2, \infty)\). These inequalities are generalizations of upper Rosenthal's inequalities for the case of the Banach space \({\mathcal L}\) with the norm \(|X |_{\mathcal L}\), \(X \in {\mathcal L}\), mentioned above.
scientific article; zbMATH DE number 825531