We consider the process $\widehatΛ_n-Λ_n$, where $Λ_n$ is a cadlag step estimator for the primitive $Λ$ of a nonincreasing function $λ$ on $[0,1]$, and $\widehatΛ_n$ is the least concave majorant of $Λ_n$. 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 $\widehatΛ_n-Λ_n$ 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 $\widehatΛ_n$ and $Λ_n$.