Title: The distance between a naive cumulative estimator and its least concave majorant,
Abstract: We consider the process $\widehat\Lambda_n-\Lambda_n$, where $\Lambda_n$ is a
cadlag step estimator for the primitive $\Lambda$ of a nonincreasing function
$\lambda$ on $[0,1]$, and $\widehat\Lambda_n$ is the least concave majorant of
$\Lambda_n$. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the
general setting considered in Durot (2007). Under this setting we prove that a
suitably scaled version of $\widehat\Lambda_n-\Lambda_n$ converges in
distribution to the corresponding process for two-sided Brownian motion with
parabolic drift and we establish a central limit theorem for the $L_p$-distance
between $\widehat\Lambda_n$ and $\Lambda_n$.