Sums of independent random variables in rearrangement invariant function spaces
type: Item
instance of: scholarly article
title: Sums of independent random variables in rearrangement invariant function spaces
zbMATH Open document ID: 0674.60051
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: The aim of the paper is to formalize one aspect of the general principle that independent random variables behave like disjoint functions. The authors' abstract runs as follows: Let X be a quasinormed rearrangement invariant function space on (0,1) which contains \(L_ q(0,1)\) for some finite q. There is an extension of X to a quasinormed rearrangement invariant function space Y on (0,\(\infty)\) so that for any sequence \((f_ i)^{\infty}_{i=1}\) of symmetric random variables on (0,1), the quasinorm of \(\sum f_ i\) in X is equivalent to the quasinorm of \(\sum \hat f_ i\) in Y, where (\^f\({}_ i)^{\infty}_{i=1}\) is a sequence of disjoint functions on (0,\(\infty)\) such that for each i, \(\hat f_ i\) has the same decreasing rearrangement as \(f_ i\). When specialized to the case \(X=L_ q(0,1)\), this result gives new information on the quantitative local structure of \(L_ q\).
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 4104115
zbMATH Keywords: invariant function space
zbMATH Keywords: rearrangement
zbMATH Keywords: decreasing rearrangement
author: William B. Johnson
author: Gideon Schechtman
reviewed by: Allan Gut
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991427
OpenAlex ID: W2049905306
DOI: 10.1214/AOP/1176991427
description: scientific article; zbMATH DE number 4104115