On deviations between empirical and quantile processes for mixing random variables

Indian Statistical Institute, Calcutta, India
Journal of Multivariate Analysis (Impact Factor: 1.06). 01/1978; DOI: 10.1016/0047-259X(78)90031-3
Source: RePEc

ABSTRACT Let {Xn} be a strictly stationary φ-mixing process with . It is shown in the paper that if X1 is uniformly distributed on the unit interval, then, for any t ∈ [0, 1], a.s. and 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 ∈ [0, 1], a.s. and sup0≤t≤1 |Fn−1(t) − t + Fn(t) − t| = 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.