Heuristic Approach to the Kolmogorov-Smirnov Theorems

Abstract

Asymptotic theorems on the difference between the (empirical) distribution function calculated from a sample and the true distribution function governing the sampling process are well known. Simple proofs of an elementary nature have been obtained for the basic theorems of Komogorov and Smirnov by Feller, but even these proofs conceal to some extent, in their emphasis on elementary methodology, the naturalness of the results (qualitatively at least), and their mutual relations. Feller suggested that the author publish his own approach (which had also been used by Kac), which does not have these disadvantages, although rather deep analysis would be necessary for its rigorous justification. The approach is therefore presented (at one critical point) as heuristic reasoning which leads to results in investigations of this kind, even though the easiest proofs may use entirely different methods. No calculations are required to obtain the qualitative results, that is the existence of limiting distributions for large samples of various measures of the discrepancy between empirical and true distribution functions. The numerical evaluation of these limiting distributions requires certain results concerning the Brownian movement stochastic process and its relation to other Gaussian processes which will be derived in the Appendix.