1. Introduction. Since the first proofs by Kolmogorov (13) and Smirnov ((14), (15)) of their well-known results on the limit distribution of the deviations of the sample distribution function, many alternative proofs of these results have been given. For example, we may cite the various approaches of Feller (4), Doob (3), Kac (8), Gnedenko and Korolyuk(7), and Anderson and Darling (1). The approaches of (3), (8) and (1) rest on a probabilistic computation regarding the Wiener process, and are justified by the paper of Donsker (2) (see also (11)). Of all these approaches, only those of (8) and (1) can be extended to obtain the limit distributions of the ‘k–sample’ generalizations of the Kolmogorov-Smirnov statistics suggested in (9), and the author ((9), (10)) and Gihman(6) carried out such proofs.