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We consider the process $\widehat\Lambda_n-\Lambda_n$, where $\Lambda_n$ is acadlag 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 Lopuha\"a (2006, 2008) to thegeneral setting considered in Durot (2007). Under this setting we prove that asuitably scaled version of $\widehat\Lambda_n-\Lambda_n$ converges indistribution to the corresponding process for two-sided Brownian motion withparabolic drift and we establish a central limit theorem for the $L_p$-distancebetween $\widehat\Lambda_n$ and $\Lambda_n$.

The distance between a naive cumulative estimator and its least concave majorant