Convergence of Reduced Empirical and Quantile Processes with Application to Functions of Order Statistics in the Non-I.I.D. Case

Abstract

Any triangular array of row independent $\mathrm{rv}$'s having continuous $\mathrm{df}$'s can be transformed naturally so that the empirical and quantile processes of the resulting $\mathrm{rv}$'s are relatively compact. Moreover, convergence (to a necessarily normal process) takes place if and only if a simple covariance function converges pointwise. Using these results we derive the asymptotic normality of linear combinations of functions of order statistics of non-i.i.d. $\mathrm{rv}$'s in the case of bounded scores.