On the almost sure topological limits of collections of local empirical processes at many different scales

Let h(n) and h(n) be two bandwidth sequences both pertaining to the domain of the strong local invariance principle, but tending to zero at different rates. We investigate the almost sure uniform clustering of Strassen type for collections of local (or increments of) empirical processes at a fixed point, under localizing scales h is an element of[h(n) h(n)]. We show that, within the framework of Strassen functional limit laws for local empirical processes, and whenever loglog(h(n)/h(n))/log log(n) -> delta > 0, the collections of all increments along bandwidths h is an element of[h(n) h(n)] almost surely admit an inner and outer topological limit. Those are Strassen balls with respective radii root delta and root 1 + delta. (C) 2015 Elsevier S.V. All rights reserved.