Characterization of the cluster set of the LIL sequence in Banach space
type: Item
instance of: scholarly article
title: Characterization of the cluster set of the LIL sequence in Banach space
zbMATH Open document ID: 0679.60006
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: A characterization of the cluster set for the sequence \(\{S_ n(2 \log \log n)^{-1/2}\}\) where \(S_ n\) is the n-th sum of i.i.d. r.v.s \(\{X_ k\}\) taking values in a separable Banach space and having mean zero and finite second moment of the norm is obtained. The cluster set is either empty or has the form \(\alpha\) K where \(\alpha\in [0,1]\) and K is the unit ball of the reproducing kernel Hilbert space associated to the covariance of \(X_ 1\). The concrete form of the cluster set is determined by convergence or divergence of a special series.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4113624
zbMATH Keywords: Banach space valued random variables
zbMATH Keywords: characterization of the cluster set
zbMATH Keywords: reproducing kernel Hilbert space
author: Kenneth S. Alexander
reviewed by: Igor S. Borisov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991424
OpenAlex ID: W2050526690
DOI: 10.1214/AOP/1176991424
description: scientific article; zbMATH DE number 4113624