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

We consider the process ${\widehat{\mathrm{\Lambda }}}_n-{\mathrm{\Lambda }}_n$, where ${\mathrm{\Lambda }}_n$ is a cadlag step estimator for the primitive $\mathrm{\Lambda }$ of a nonincreasing function $\lambda$ on $[0,1]$, and ${\widehat{\mathrm{\Lambda }}}_n$ is the least concave majorant of ${\mathrm{\Lambda }}_n$. We extend the results in Kulikov and Lopuha"a (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of ${\widehat{\mathrm{\Lambda }}}_n-{\mathrm{\Lambda }}_n$ 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 ${\widehat{\mathrm{\Lambda }}}_n$ and ${\mathrm{\Lambda }}_n$.