The law of the iterated logarithm for empirical processes on Vapnik-Červonenkis classes

AbstractNecessary and sufficient conditions are established for the bounded and compact laws of the iterated logarithm for empirical processes indexed by classes of functions which have the Vapnik-Červonenkis property. Results of Ledoux and Talagrand (Ann. Probab. 16 1242–1264, and in press) reduce the problem fromone of almostsure behavior to one of in-probability behavior. The special case of weighted empirical processes indexed by sets is considered