On deviations between empirical and quantile processes for mixing random variables

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 X₁ 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.