On functional versions of the arc-sine law
type: Item
instance of: scholarly article
title: On functional versions of the arc-sine law
zbMATH Open document ID: 1201.60030
author: István Berkes
author: Lajos Horváth
author: Siegfried Hörmann
published in: Journal of Theoretical Probability
publication date: 2010-04-23T00:00:00Z
review text: Suppose \(R(t)\) and \(R_n(t)\), \(n\geq 1\), are random processes in \(D(0,1)\) such that \(R_n(t)\) converges weakly to \(R(t)\) in \(D(0,1)\). For \(z\in(-\infty,\infty)\) define \(\Lambda_n(z)= \int^1_0\mathbf{1}\{R_n(t)\leq z\}\,dt\) and similarly \(\Lambda(z)\) for the process \(R(t)\). Under some additional conditions on path properties of the limit proces \(R(t)\) it is proved that \(\Lambda_n(z)\) converges weakly to \(\Lambda(z)\) in \(D[-\infty,\infty]\). This general theorem contains the following important example. Let \(X_1, X_2,\dots\) are i.i.d. random variables with mean 0 and the unit variance. Then \[ {1\over n}\sum^n_{k=1}\mathbf{1}\{(X_1,+\cdots+ X_k)/\sqrt{k}\leq z\} \] converges weakly in \(D[-\infty,\infty]\) to \(\int^1_0 \mathbf{1}\{W(t)/\sqrt{t}\leq z\}\,dt\), where \(W(t)\) is a standard Brownian motion. Taking \(z= 0\) one obtains the classical Lévy's first arc-sine law. The general theorem is used to derive several convergence results for normalized partial sums of dependent random variables both for Gaussian and normalized Lévy limit processes \(R(t)\) as well as for extreme-value statistics of i.i.d. random variables.
reviewed by: T. Inglot
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60G52
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 5700610
zbMATH Keywords: arc-sine law
zbMATH Keywords: weak invariance principle
zbMATH Keywords: partial sums
zbMATH Keywords: extreme-value statistics
MaRDI profile type: Publication
full work available at URL: s10959-008-0181-7
OpenAlex ID: W2169480853
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DOI: 10.1007/S10959-008-0181-7
description: scientific article; zbMATH DE number 5700610