Functional laws of the iterated logarithm for large increments of empirical and quantile processes

Let {[alpha]n(t),0[less-than-or-equals, slant]t[less-than-or-equals, slant]1} and {[beta]n(t),0[less-than-or-equals, slant]t[less-than-or-equals, slant]1} be the empirical and quantile processes generated by the first n observations from an i.i.d. sequence of random variables with a uniform distribution on (0, 1). Let 0 < hn < 1 be a sequence of constants such that hn-->0 and (log(1/hn))/log log n --> c [epsilon][0,[infinity]) as n-->[infinity]. Under suitable additional regularity conditions imposed upon hn, we prove functional laws of the iterated logarithm for . We present applications of these results to nonparametric densityestimation, and prove a conjecture of Shorack and Wellner (1986) concerning the limiting behaviour of the maximal increments of [alpha]n and [beta]n.