On Weak Convergence of Stochastic Processes with Multidimensional Time Parameter

Abstract

The well-known space $D\lbrack 0, 1\rbrack$ is generalized to $k$ time dimensions and some properties of this space $D_k$ are derived. Then, following the "classical" lines as presented in Billingsley [1], a Skorohod-metric, tightness criteria and some other results concerning weak convergence are given. The theory is applied to prove weak convergence of two generalizations of the one-dimensional empirical process and of the Kolmogorov-Smirnov test statistic of independence. Stochastic processes with multidimensional time parameter and their weak convergence have been investigated by several authors. Dudley [4] established a theory of convergence of stochastic processes with sample functions in nonseparable metric spaces. Later on, Wichura [11] (see also Wichura [12]) modified the concepts of Dudley and developed them systematically. He applied his theory to a space which is with minor changes our space $D_k$. Weak convergence in the sense of Wichura [12] and ours differ usually, but both concepts coincide if the limit process has--with probability one--continuous sample functions only. From here it follows that the results of Dudley and Wichura concerning weak convergence of multivariate empirical processes are equivalent to ours. At least two further authors proved the convergence of multivariate empirical processes, namely LeCam [8] and Bickel [1]. Our proof follows the classical approach of Parthasarathy [9] using an argument of Kuelbs [7] to carry over the proof from 1 to $k$ dimensions. Kuelbs however deals properly with the "interpolated sum" process for two-dimensional time parameter. The space $D_k$ seems to be defined for the first time in connection with multivariate processes by Winkler [13], yet his investigations are not concerned with weak convergence. Another generalization of the space $D\lbrack 0, 1\rbrack$ and the Skorohod metric to functions on more general spaces than $E_k$ is given in the paper [10] of Straf, in which there are applications to genuinely discontinuous limit processes.