The rank of the present excursion
type: Item
instance of: scholarly article
title: The rank of the present excursion
zbMATH Open document ID: 0819.60069
DOI: 10.1016/0304-4149(95)91544-B
author: Carel L. Scheffer
published in: Stochastic Processes and their Applications
publication date: 1995-08-15T00:00:00Z
review text: One way to view standard Brownian motion is as a succession of excursions from 0; indexed by local time, these excursions form a Poisson point process. About ten years ago the reviewer tried to internalize this important result by finding the probability that the excursion straddling a given time instant \(t\) is of record duration (a moderate understanding of Brownian motion suffices to show that this probability does not depend on \(t\), i.e. is `a universal constant'). Other, more urgent things had to be done, but the problem continued to exist as the reviewer's favourite `coffee-room problem', until it was finally solved by Scheffer in the paper under review. For his solution Scheffer employs an impressive array of techniques, especially from the theory of point processes. An application is also given to renewal processes with heavy-tailed lifetime distributions. This is a very nice paper, using sophisticated theories to achieve surprisingly explicit results; the universal constant, for example, is 0.80031... It was written, accepted and published around the time of the author's retirement, so it has some additional meta-value as a(nother) counterexample to one of the mathematician's favourite coffee-room myths.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 735475
zbMATH Keywords: Brownian motion
zbMATH Keywords: excursions
zbMATH Keywords: Palm probabilities
zbMATH Keywords: heavy-tailed distributions
zbMATH Keywords: local time
zbMATH Keywords: point processes
reviewed by: Rudolf Grübel
MaRDI profile type: Publication
cites work: Q5583383
cites work: Q5633348
cites work: Q5532811
cites work: The rank of the current lifetime
cites work: Q4052400
cites work: Q3771297
description: scientific article; zbMATH DE number 735475