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November, 1983 The Concave Majorant of Brownian Motion
Piet Groeneboom
Ann. Probab. 11(4): 1016-1027 (November, 1983). DOI: 10.1214/aop/1176993450


Let $S_t$ be a version of the slope at time $t$ of the concave majorant of Brownian motion on $\lbrack 0, \infty)$. It is shown that the process $S = \{1/S_t: t > 0\}$ is the inverse of a pure jump process with independent nonstationary increments and that Brownian motion can be generated by the latter process and Brownian excursions between values of the process at successive jump times. As an application the limiting distribution of the $L_2$-norm of the slope of the concave majorant of the empirical process is derived.


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Piet Groeneboom. "The Concave Majorant of Brownian Motion." Ann. Probab. 11 (4) 1016 - 1027, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0523.60079
MathSciNet: MR714964
Digital Object Identifier: 10.1214/aop/1176993450

Primary: 60J75
Secondary: 60J75 , 62E20

Keywords: Brownian excursions , Brownian motion , concave majorant , convex minorant , empirical process , limit theorems , slope process

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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