On deviations between empirical and quantile processes for mixing random variables

Abstract

Let { X n } be a strictly stationary φ-mixing process with Σ j=1 ∞ φ 1 2 (j) < ∞ . It is shown in the paper that if X 1 is uniformly distributed on the unit interval, then, for any t ∈ [0, 1], |F n −1(t) − t + F n(t) − t| = O(n − 3 4 ( log log n) 3 4 ) a.s. and sup 0≤t≤1 |F n −1(t) − t + F n(t) − t| = (O(n − 3 4 ( log n) 1 2 ( log log n) 1 4 ) a.s., where F n and F n −1( t) denote the sample distribution function and tth sample quantile, respectively. In case { X n } is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t ∈ [0, 1], |F n −1(t) − t + F n(t) − t| = O(n − 3 4 ( log n) 1 2 ( log log n) 3 4 ) a.s. and sup 0≤ t≤1 | F n −1( t) − t + F n ( 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 φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.