Laws of the iterated logarithm for partial sum processes indexed by functions
type: Item
instance of: scholarly article
title: Laws of the iterated logarithm for partial sum processes indexed by functions
zbMATH Open document ID: 0671.60021
DOI: 10.1007/BF01054022
author: Michael T. Lacey
published in: Journal of Theoretical Probability
publication date: 1989-01-01T00:00:00Z
review text: For \(d\in N\), let \(\{X_ j\), \(j\in N^ d\}\) be an array of independent copies of a random variable X with E X\(=0\), E \(X^ 2=1\). For any Borel set \(A\subset [0,1)^ d=I\) define the random measure \(S_ n(A,X)=\sum_{j\in nA}X_ j\). Further if f is a measurable function on I one can define \[ S_ n(f,X)=\int_{I}f(t)S_ n(dt,X). \] We say that X satisfies the bounded law of the iterated logarithm for a class F of measurable functions if \[ \{(2n \log \log n)^{-1/2}S_ n(f,X)\} \] is a bounded subset of \(\ell^{\infty}(F)\). The notion of the compact LIL can be defined similarly. The author gives conditions on F implying that X satisfies the bounded resp. the compact LIL. Note that if \(F=\{1_{[0,t)}\), \(0\leq t\leq 1\}\) we obtain the classical LIL as special case.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 4098393
zbMATH Keywords: partial sum processes
zbMATH Keywords: metric entropy
zbMATH Keywords: law of the iterated logarithm
reviewed by: Pál Révész
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4098393