Inequalities for sums of independent random variables in Lorentz spaces
type: Item
instance of: scholarly article
title: Inequalities for sums of independent random variables in Lorentz spaces
zbMATH Open document ID: 1342.46033
DOI: 10.1216/RMJ-2015-45-5-1631
author: Ghadir Sadeghi
published in: Rocky Mountain Journal of Mathematics
publication date: 2016-02-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1453817257
review text: 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 \(L^{p,q}\).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 6538962
zbMATH Keywords: Khinchin inequality
zbMATH Keywords: random variable
zbMATH Keywords: Rosenthal inequality
zbMATH Keywords: Lorentz space
zbMATH Keywords: Lorentz-Zygmund space
zbMATH Keywords: interpolation
reviewed by: Vakhtang V. Kvaratskhelia
MaRDI profile type: Publication
cites work: Inequalities for Sums of Independent Random Variables
cites work: Weighted Lorentz spaces and the Hardy operator
cites work: A function parameter in connection with interpolation of Banach spaces.
cites work: An extension of Rosenthal's inequality
cites work: Sums of independent random variables in rearrangement invariant function spaces
cites work: Compact and strictly singular operators on certain function spaces
cites work: Q4188284
cites work: Q3336223
cites work: Interpolation with a parameter function.
cites work: On the subspaces of \(L^p\) \((p > 2)\) spanned by sequences of independent random variables
cites work: Spaces \(\Lambda_\alpha\)(X) and interpolation
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description: scientific article; zbMATH DE number 6538962