Laws of the iterated logarithm in the tails for weighted uniform empirical processes

Characterizations of laws of the iterated logarithm for the supremum of weighted uniform $\lbrack 0, 1\rbrack^d$ empirical processes taken over increasingly smaller regions near the origin are obtained. These results have proven to be a valuable tool in the derivation of laws of the iterated logarithm for sums of extreme values. They also constitute a further continuation of the study of the almost sure behavior of weighted uniform empirical processes, which in a certain sense was begun by Csaki.