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.
John H. J. Einmahl. David M. Mason. "Laws of the Iterated Logarithm in the Tails for Weighted Uniform Empirical Processes." Ann. Probab. 16 (1) 126 - 141, January, 1988. https://doi.org/10.1214/aop/1176991889