Inequalities for sums of independent random variables in Lorentz spaces
Let \(\{r_k\}_{k\geq1}\) be a Rademacher sequence on a probability space \((\Omega,\mathcal{F},\text{P})\). The classical Khinchin inequalities assert that there exist some positive constants \(c_1\) and \(c_2\) such that, for any finite sequence of scalars \(\{\alpha_k\}\) and any number \(0<p<\infty\), we have \[ c_1\left(\sum_k|\alpha_k|^2\right)^{1/2}\leq\left(\int_{\Omega}\left|\sum_k\alpha_k r_k(\omega)\right|^p\right)^{1/p} \leq c_2\left(\sum_k|\alpha_k|^2\right)^{1/2}. \] \textit{H. P. Rosenthal} [Isr. J. Math. 8, 273--303 (1970; Zbl 0213.19303)] generalized the Khinchin inequality by replacing \(\{r_k\}_{k\geq1}\) with an arbitrary sequence \(\{X_k\}_{k\geq1}\) of independent symmetric random variables on a probability space \((\Omega,\mathcal{F},\text{P})\) and \(\{X_k\}_{k\geq1}\subset L^p(\Omega)\), \(p\geq2\). In this paper, by using interpolation with a function parameter, a moment inequality is proved for sums of independent random variables in Lorentz spaces \(\Lambda^p(\varphi)\). These estimates generalize Rosenthal's inequalities in the Lorentz-Zygmund spaces \(L^{p,q}(\log L)^\gamma\) as well as in the Lorentz spaces \(L^{p,q}\).
scientific article; zbMATH DE number 6538962
scientific article