On nonincrease of Brownian motion
type: Item
instance of: scholarly article
title: On nonincrease of Brownian motion
zbMATH Open document ID: 0719.60086
author: K. Burdzy
published in: The Annals of Probability
publication date: 1990-01-01T00:00:00Z
review text: The author's opening paragraph states the contents of this note: \textit{D. Aldous} pointed out in his recent book [Probability approximations via the Poisson clumping heuristic. New York etc.: Springer-Verlag (1989; Zbl 0679.60013), K7, K13] that none of the published proofs of the nonincrease of Brownian motion is totally satisfactory. The original proof of \textit{A. Dvoretzky}, \textit{P. Erdős} and \textit{S. Kakutani} [Proc. 4th Berkeley Symp. Math. Stat. Probab. 2, 103--116 (1961; Zbl 0111.15002)] is hard; those of \textit{O. Adelman} [Isr. J. Math. 50, 189--192 (1985; Zbl 0573.60030)], \textit{I. Karatzas} and \textit{S. E. Shreve} [Brownian motion and stochastic calculus. New York etc.: Springer-Verlag (1988; Zbl 0638.60065)] or \textit{F. B. Knight} [Essentials of Brownian motion and diffusion. Providence, R.I.: AMS (1981; Zbl 0458.60002)] do not shed much light on the difference between points of increase and local maxima. Aldous' own argument is only a ``Poisson clumping heuristic''. This note is an attempt at the impossible, i.e., a proof which is at the same time rigorous, short, simple, elementary, and last but not least, elucidates the difference between the points of increase and local maxima. I believe that the proof is new, although its ideas may be traced back to Adelman (loc. cit.), Aldous (loc. cit.) and \textit{B. Davis} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 359--367 (1983; Zbl 0506.60078)].
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 4186800
zbMATH Keywords: points of increase
zbMATH Keywords: local maxima
reviewed by: Michael Craig Cranston
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990732
OpenAlex ID: W1969051071
DOI: 10.1214/AOP/1176990732
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description: scientific article