The Rosenthal inequality in symmetric spaces (Q1814350)

Let \(X\) be a real random variable on \((\Omega,P)\), and let \(F(x)=P[X>x]\), \(x\in R^ 1\). Let the random variable whose characteristic function is \[ f(t)=\exp[\int^ \infty_{-\infty}(e^{itx}-1)dF(x)] \] be denoted by \(Y_ F\). For a rearrangement invariant space \(E\) the author calls \(E\) to possess the Kruglov property if the inclusions \(X\in E\) and \(Y_ F\in E\) are equivalent. The present paper investigates this property in connection with so-called Bahr-Essen-, Rosenthal-, and weak Rosenthal properties of rearrangement invariant spaces, and also deals with upper \(p\)-estimates in and Boyd indices of the latter [cf. \textit{J. Lindenstrauss}, \textit{L. Tzafrizi}, Classical Banach spaces. II (1979; Zbl 0403.46022)]. The obtained results are applied to Lorentz spaces \(L_{p,q}\).