The Bounded Law of the Iterated Logarithm for the Weighted Empirical Distribution Process in the Non-I.I.D. Case

Abstract

Using a simple symmetrization procedure, an upper bound is obtained for the probability distribution of various kinds of weighted empirical distribution processes where the underlying real valued random variables are not identically distributed. These probability bounds are used to obtain bounded laws of the iterated logarithm for empirical processes with different kinds of weighting. They are also used to obtain a one sided version of Daniel's theorem in the non-i.i.d. case.