On the law of large numbers in 2-uniformly smooth Banach spaces
type: Item
instance of: scholarly article
title: On the law of large numbers in 2-uniformly smooth Banach spaces
zbMATH Open document ID: 0545.60015
author: Bernard Heinkel
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: Kolmogorov's strong law of large numbers is extended to random variables which take their values in a 2-uniformly smooth Banach space \((B,\| \cdot \|)\). The result reads as follows: Let \((X_ k\), \(k\in {\mathbb{N}})\) be a sequence of centered independent random variables taking values in \((B,\| \cdot \|)\). Suppose that the following conditions hold: 1) There exists a positive constant k such that: \(\| X_ n\| \leq kn/\sqrt{L_ 2n}^ a.\)s., all \(n\in {\mathbb{N}}\), where \(L_ 2n=\log(\log(\sup \{n,e^ e\}));\quad 2)\quad \lim_{n\to \infty}(\sum^{n}_{j=1}E\| X_ j\|^ 2)/n^ 2=0;\quad 3)\quad \sum^{\infty}_{j=1}\sup_{\| f\|_{B'}\leq 1}(Ef^ 2(X_ j))/j^ 2<+\infty.\) Then the sequence \((X_ k\), \(k\in {\mathbb{N}})\) satisfies the strong law of large numbers.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3868316
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: uniformly smooth Banach space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176993233
OpenAlex ID: W1980376756
DOI: 10.1214/AOP/1176993233
description: scientific article; zbMATH DE number 3868316