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(t0)>0, f′(t0)<0, and f′ is continuous in a neighborhood of t0, then blalim supn→∞(n2loglogn)1/3(f̂n(t0)−f(t0))=|f(t0)f′(t0)/2|1/32M almost surely where M≡supg∈GTg=(3/4)1/3andTg≡argmaxu{g(u)−u2}; here G is the two-sided Strassen limit set on 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; see Strassen  [26].