## The Annals of Probability

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

### The Concave Majorant of Brownian Motion

#### 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

**Permanent link to this document**

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

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176993450

**Mathematical Reviews number (MathSciNet)**

MR714964

**Zentralblatt MATH identifier**

0523.60079

**Subjects**

Primary: 60J75: Jump processes

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

**Keywords**

Concave majorant convex minorant slope process Brownian motion Brownian excursions empirical process limit theorems

#### Citation

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