FUNCTIONAL LAWS OF THE ITERATED LOGARITHM FOR THE INCREMENTS OF EMPIRICAL AND QUANTILE PROCESSES

Let {alpha(n)(t), 0 less-than-or-equal-to t less-than-or-equal-to 1} and {beta(n)(t), 0 less-than-or-equal-to t less-than-or-equal-to 1} be the empirical and quantile processes generated by the first n observations from an i.i.d. sequence of uniformly distributed random variables on (0, 1). Let 0 < alpha(n) < 1 be a sequence of constants such that alpha(n) --> 0 as n --> infinity. We investigate the strong limiting behavior as n --> infinity of the increment functions {alpha(n)(t + alpha(n)s) - alpha(n)(t), 0 less-than-or-equal-to s less-than-or-equal-to 1} and {beta(n)(t + alpha(n)s) - beta(n)(t), 0 less-than-or-equal-to s less-than-or-equal-to 1}, where 0 less-than-or-equal-to t less-than-or-equal-to 1 - alpha(n). Under suitable regularity assumptions imposed upon alpha(n), we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.