The rank of the present excursion
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.
scientific article; zbMATH DE number 735475