Large-sample results for Kolmogorov-Smirnov statistics for discrete distributions

The asymptotic distributions of the Kolmogorov-Smirnov goodness-of-fit statistics are considered when the hypothesized distribution is discrete. Each of the statistics is shown to have the same distribution as a continuous functional of an associated empirical process on the unit interval. Using known weak convergence properties for the empirical process, the asymptotic distributions of the Kolmogorov-Smirnov statistics are derived. A discussion and example are included.