Clustering behavior of finite variance partial sum processes (Q1898839)

The authors obtain clustering rates in Strassen's functional law of the iterated logarithm for finite variance partial sum processes in one dimension. They also obtain a general characterization of these rates which enables them in matching the rates of Brownian motion for a partial sum process at points \(f\) inside the Strassen set \(K\) and also at points on the boundary of \(K\). Somewhat surprisingly the results are obtained under relatively mild moment conditions on a partial sum process. The proof is based on a strong approximation result that is particularly efficient for large excursions of normalized partial sum processes.