## The Annals of Probability

### The Concave Majorant of Brownian Motion

Piet Groeneboom

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 11, Number 4 (1983), 1016-1027.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993450

Digital Object Identifier
doi:10.1214/aop/1176993450

Mathematical Reviews number (MathSciNet)
MR714964

Zentralblatt MATH identifier
0523.60079

JSTOR

Subjects
Primary: 60J75: Jump processes
Secondary: 60J75: Jump processes 62E20: Asymptotic distribution theory

#### Citation

Groeneboom, Piet. The Concave Majorant of Brownian Motion. Ann. Probab. 11 (1983), no. 4, 1016--1027. doi:10.1214/aop/1176993450. https://projecteuclid.org/euclid.aop/1176993450