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

Let hnhn and hnhn 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∈[hn,hn]h∈[hn,hn]. We show that, within the framework of Strassen functional limit laws for local empirical processes, and whenever loglog(hn/hn)/loglog(n)→δ>0loglog(hn/hn)/loglog(n)→δ>0, the collections of all increments along bandwidths h∈[hn,hn]h∈[hn,hn] almost surely admit an inner and outer topological limit. Those are Strassen balls with respective radii δ and 1+δ.