Lim inf results for Gaussian samples and Chung's functional LIL
type: Item
instance of: scholarly article
title: Lim inf results for Gaussian samples and Chung's functional LIL
zbMATH Open document ID: 0849.60022
author: Wenbo V. Li
author: Michel Talagrand
author: Jim Kuelbs
published in: The Annals of Probability
publication date: 1996-10-29T00:00:00Z
review text: Let \(X, X_1, X_2, \dots\) be i.i.d. centered Gaussian random elements of a Banach space \(B\) with common law \(\mu\) and let \(\psi (\varepsilon) = \log \mu (x : |x |\leq \varepsilon)\). The authors study \[ \liminf_{n \to \infty} d(n)^{-1} |X_n/(2 \log n)^{1/2} - f | \] for elements \(f\) of \(B\) where \(d = d(n)\) is a solution of certain transcendental equations involving \(\psi (d(2 \log n)^{1/2})\) and in addition \(f\) when \(|f |_\mu = 1\); the results differ according to whether \(|f |_\mu < 1\) or \(|f |_\mu = 1\). Conditions are also given for \(d_1 (n) \approx d_2 (n)\) when \(|f_1 |_\mu = |f_2 |_\mu = 1\), and the exact constants are obtained for some examples in Hilbert space.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 828370
zbMATH Keywords: Banach space
zbMATH Keywords: Gaussian samples
zbMATH Keywords: functional iterated logarithm law
zbMATH Keywords: Wiener measure
reviewed by: Søren Asmussen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988488
OpenAlex ID: W2041640202
DOI: 10.1214/AOP/1176988488
description: scientific article; zbMATH DE number 828370