Rate of convergence in the weak law of large numbers in a Banach space
type: Item
instance of: scholarly article
title: Rate of convergence in the weak law of large numbers in a Banach space
zbMATH Open document ID: 0598.60014
DOI: 10.1007/BF00970334
author: B. George
published in: Lithuanian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let \((B,\| \cdot \|)\) be a real separable Banach space, \(L_ 0(B)=L_ 0(\Omega,{\mathcal F},P;B)\) be the set of all B-valued random variables, defined on the probability space (\(\Omega\),\({\mathcal F},P)\), \(X_ 1,...,X_ n\) be independent copies of the B-valued random variable X, \(S_ n(X)=X_ 1+...+X_ n\); EX denote the Bochner mean. Let the numbers p,q be real, the space ''of weak type q'' is \[ \Lambda^ q_ 0(B)=\{X\in L_ 0(B),\quad EX=0\quad and\quad \lim_{t\to 0}t^ qP(\| X\| >t)=0\}, \] and \[ RLLN_{p,q}(B)=\{X\in L_ 0(B),\quad for\quad each\quad \epsilon >0,\quad P(\| S_ n(X)\| >\epsilon n^{1/p})=o(n^{1-q})\}. \] In this paper the author defines type (p,r,q) as follows: Let \(1\leq p<2\), \(p\leq q\leq r\leq \infty\). A Banach space is called a space of type (p,r,q), if there exists a constant \(c>0\) such that for any finite collection \((x_ i)^ n_{i=1}\subset B\), \[ E\| \sum^{n}_{1}r_ ix_ i\| \leq cn^{1/p-1/q}\|| (x_ i)\||_{q,r}, \] where \(\|| (x_ i)\||_{q,r}\) denotes the Lorentz norm. Denote by \({\mathcal D}(p,r,q)\) the class of Banach spaces of type (p,r,q). The main theorem is that: if \(1\leq p<2\), \(1<q<\infty\), then \(\Lambda_ 0^{pq}(B)\subset RLLN(B)\) iff the Banach space B has type (p,\(\infty,pq)\).
reviewed by: Xin-Tai Yu
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3965070
zbMATH Keywords: rate of convergence
zbMATH Keywords: law of large numbers
zbMATH Keywords: geometry of the Banach space
zbMATH Keywords: Rademacher sequence
zbMATH Keywords: Banach space of stable type
MaRDI profile type: Publication
Wikidata QID: Q114852768
cites work: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach
cites work: Remarks on stable measures on Banach spaces
cites work: On the type and cotype of Banach spaces
cites work: Sums of independent Banach space valued random variables
cites work: Q3968218
cites work: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes
cites work: Q4748883
cites work: Inequalities for B-valued random vectors with applications to the strong law of large numbers
cites work: Existence and Convergence of Probability Measures in Banach Spaces
cites work: Convergence Rates in the Law of Large Numbers
cites work: Q5524996
cites work: Central limit problem for symmetric case: Convergence to non-Gaussian laws
cites work: Stable Measures and Central Limit Theorems in Spaces of Stable Type
cites work: Rates of convergence for the laws of large numbers for independent Banach-valued random variables
cites work: Q3918793
cites work: Q5667036
full work available at URL: https://doi.org/10.1007/bf00970334
OpenAlex ID: W2033613086
description: scientific article; zbMATH DE number 3965070