The general form of the law of the iterated logarithm for generalized Kolmogorov-Smirnov-Renyi statistics

One result of Smirnov's important paper [Uspehi Mat. Nauk. 10, 179–206, (in Russian)] yields exponential bounds for the large deviations of his one-sided Smirnov statistic and the two-sided Kolmogorov statistic. In the present paper exponential bounds are given for the large deviations of a wide class of Kolmogorov-Smirnov-Renyi type statistics. As a by-product, exponential bounds for the large deviations of the corresponding limit distributions are obtained.