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 $[0,\mathrm{\infty })$. 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.