Groeneboom introduced a jump process that can be used (among other things) to study the asymptotic properties of the Grenander estimator of a monotone density. In this paper we derive the asymptotic normality of a suitably rescaled version of the $L_1$ error of the Grenander estimator, using properties of this jump process.
"Asymptotic normality of the $L_1$ error of the Grenander estimator." Ann. Statist. 27 (4) 1316 - 1347, August 1999. https://doi.org/10.1214/aos/1017938928