Bootstraps of sums of independent but not identically distributed stochastic processes

Abstract

A central limit theorem is developed for sums of independent but not identically distributed stochastic processes multiplied by independent real random variables with mean zero. Weak convergence of the Hoffmann–J ørgensen–Dudley type, as described in van der Vaart and Wellner (Weak Convergence and Empirical Processes, Springer, New York, 1996), is utilized. These results allow Monte Carlo estimation of limiting probability measures obtained from application of Pollard's (Empirical Processes: Theory and Applications, IMS, Hayward, CA, 1990) functional central limit theorem for empirical processes. An application of this theory to the two-parameter Cox score process with staggered entry data is given for illustration. For this process, the proposed multiplier bootstrap appears to be the first successful method for estimating the associated limiting distribution. The results of this paper compliment previous bootstrap and multiplier central limit theorems for independent and identically distributed empirical processes.