A law of the iterated logarithm for Grenander's estimator

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0)>0$, $f’(t_0) < 0$,and$f’$iscontinuousinaneighborhoodof$t_0$, then \begin{eqnarray*} \limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} ( \widehat{f}n (t_0 ) - f(t_0) ) = \left| f(t_0) f’(t_0)/2 \right|^{1/3} 2M \end{eqnarray*} almost surely where $ M \equiv \sup{g \in {\cal G}} T_g = (3/4)^{1/3}$and$ T_g \equiv \mbox{argmax}_u { g(u) - u^2 } $;here${\cal G}$isthetwo-sidedStrassenlimitseton$R$. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom’s switching relation, and properties of Strassen’s limit set analogous to distributional properties of Brownian motion.