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

Any triangular array of row independent rv's having continuous df's can be transformed naturally so that the empirical and quantile processes of the resulting 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. rv's in the case of bounded scores.