Asymptotic Representations of the Multivariate Empirical Distribution Function and Applications

Often, many complicated statistics used as estimators or test statistics take the form of the (multivariate) empirical distribution function evaluated at a random vector (V_n). Denote such statistics by S_n. This paper describes methods for the study of various asymptotic properties of S_n. First, under minimal assumptions, a weak asymptotic representation for S_n is derived. This result may be used to show the asymptotic normality of S_n. Second, under slightly more stringent regularity conditions, an almost sure representation of S_n, with suitable order (as.) of the remainder term is studied and then a law of the iterated logarithm for S_n, is derived. In this context, strong convergence results from a sequential point of view are also studied. Finally, weak convergence to a Brownian motion process is established. As an application, we show the limiting normality of S_n, for a random number of summands.