Strong Approximation of the Quantile Processes and Its Applications under Strong Mixing Properties

Given some regularity conditions on the distribution F(·) of a random X1, ..., Xn emanating from a strictly stationary sequence of random variables satisfying a strong mixing condition, it is shown that the sequence of quantile processes {nf(F-1(s))(F-1n(s) - F-1(s)); 0 < s < 1} behaves like a sequence of Brownian bridges {Bn(s); 0<s<1}. The latter is then utilized to construct (i) simultaneous bounds for the unknown quantile function F-1(s), and (ii) a tolerance interval for predicting a future observation. Some numerical investigations of the results are also discussed.