A law of the iterated logarithm for Grenanders 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 $f$ is continuous in a neighborhood of $t_0$, then begin{eqnarray*} limsup_{nrightarrow infty} left ( frac{n}{2log 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 ${cal G}$ is the two-sided Strassen limit set on $R$. The proof relies on laws of the iterated logarithm for local empirical processes, Groenebooms switching relation, and properties of Strassens limit set analogous to distributional properties of Brownian motion.