The LIL when X is in the domain of attraction of a Gaussian law
type: Item
instance of: scholarly article
title: The LIL when X is in the domain of attraction of a Gaussian law
zbMATH Open document ID: 0572.60010
published in: The Annals of Probability
publication date: 1985-01-01T00:00:00Z
review text: A mean zero random variable X with values in a Banach space B in the domain of normal attraction of a Gaussian law satisfies the compact law of the iterated logarithm if and only if \(E(\| X\|^ 2/L_ 2\| X\|)<\infty\). In the paper under review, the author mainly shows that an analogous result can be obtained when X is merely in the domain of attraction of a Gaussian law Z, i.e. when there exist a sequence \(d_ n\nearrow \infty\) and \(\{\delta_ n\}\) in B such that \((S_ n-\delta_ n)/d_ n\) converges in distribution to Z. Under this assumption, the author actually proves the existence of smooth coefficients \(d_ n\) which leads to a law of the iterated logarithm with regularity properties of the normalizing sequence. The corresponding functional law of the iterated logarithm is obtained in this setting and the clustering phenomena are examined with great care: in contrast with the situation of normal attraction, when X is in the domain of attraction of Z, the unit ball of the R.K.H.S. of Z may be a proper subset of the cluster set of the law of the iterated logarithm.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 3913334
zbMATH Keywords: domain of normal attraction
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: functional law of the iterated logarithm
zbMATH Keywords: clustering phenomena
zbMATH Keywords: cluster set of the law of the iterated logarithm
author: Jim Kuelbs
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992910
OpenAlex ID: W1994988462
DOI: 10.1214/AOP/1176992910
description: scientific article; zbMATH DE number 3913334