On deviations between empirical and quantile processes for mixing random variables

Let {Xn} be a strictly stationary [phi]-mixing process with [Sigma]j=1[infinity] [phi]1/2(j) < [infinity]. It is shown in the paper that if X1 is uniformly distributed on the unit interval, then, for any t [set membership, variant] [0, 1], Fn-1(t) - t + Fn(t) - t = O(n-3/4(log log n)3/4) a.s. and sup0<=t<=1 Fn-1(t) - t + Fn(t) - t = (O(n-3/4(log n)1/2(log log n)1/4) a.s., where Fn and Fn-1(t) denote the sample distribution function and tth sample quantile, respectively. In case {Xn} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [set membership, variant] [0, 1], Fn-1(t) - t + Fn(t) - t = O(n-3/4(log n)1/2(log log n)3/4) a.s. and sup0<=t<=1 Fn-1(t) - t + Fn(t) - t = O(n-3/4(log n)(log log n)1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for [phi]-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.