The arcsine law
type: Item
instance of: scholarly article
title: The arcsine law
zbMATH Open document ID: 0921.60021
DOI: 10.1023/A:1021748727681
author: Jørgen Hoffmann-Jørgensen
published in: Journal of Theoretical Probability
publication date: 1999-09-29T00:00:00Z
review text: The classical equivalence principle states that the number of positive sums in the first \(n\) trials in a random walk and the first time the maximal partial sum in the first \(n\) trials have the same distribution, and the classical arcsine law [see \textit{W. Feller}, ``An introduction to probability theory and its applications. Vol. II'' (1966; Zbl 0138.10207)] gives a necessary and sufficient condition for the above two random variables after dividing by \(n\) to converge in law to the same arcsine distribution. The classical proofs usually apply advanced methods from complex analysis and Fourier analysis together with some sort of Tauber theorems and tricky and ingenious combinatorial methods. With the help of a simple and elementary extension of the Cesàro lemma the author provides a proof of the equivalence principle for a general class of integer-valued random vectors of the two concerned random variables. The author also provides examples of arcsine sequences which arise naturally in Markov chains, last visiting times in Markov chains, ladder points and occurence numbers, and recurrence times in \textit{J. F. C. Kingman}'s ``Regenerative phenomena'' (1972; Zbl 0236.60040)].
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 1275650
zbMATH Keywords: random walk
zbMATH Keywords: arcsine
zbMATH Keywords: classical equivalence principle
reviewed by: Adhir K. Basu
MaRDI profile type: Publication
full work available at URL: a:1021748727681
OpenAlex ID: W35473516
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description: scientific article; zbMATH DE number 1275650