SOME LAW OF THE ITERATED LOGARITHM TYPE RESULTS FOR THE EMPIRICAL PROCESS

We consider Z^±_n= sup_{0< t ≤ 1/2}2 U^±_n (t)/(t(1- t))^1/2, where + and -denote the positive and negative parts respectively of the sample paths of the empirical process U_n. U^±_n and U_n are seen to behave rather differently, which is tied to the asymmetry of the binomial distribution, or to the asymmetry of the distribution of small order statistics. Csáki (1975) showed that log Z^±_n/log₂n is the appropriate normalization for a law of the iterated logarithm (LIL) for Z^±_n we show that Z^-_n/(2 log₂n)^1/2 is the appropriate normalization for Z^-_n. Csörgö & Révész (1975) posed the question: if we replace the sup over (0,1/2) above, by -the sup over a_n, 1-a_n] where a_n→0, how fast can a_n→0 and still have |Z_n|/(2 log₂n)^1/2 maintain a finite lim sup a.s.? This question is answered herein. The techniques developed are then used in Section 4 to give an interesting new proof of the upper class half of a result of Chung (1949) for |U_n(t)|. The proofs draw heavily on James (1975); two basic inequalities of that paper are strengthened to their potential, and are felt to be of independent interest.