This paper is devoted to some problems related to asymptotic power and efficiency of tests of Kolmogorov-Smirnov type. The results are derived by means of the theory of stochastic processes. The concepts of local asymptotic power and local asymptotic efficiency of tests are introduced according to [7]. Section 4 contains the evaluation of the local asymptotic power of one-sided tests of Kolmogorov-Smirnov type and of Renyi's one-sided test. The asymptotic power of these tests may be expressed by series expansion. Some theoretical results related to terms of the series are given and used for approximations and bounds of the asymptotic power. The paper is an excerpt from the author's dissertation written with the guidance of Professor Jaroslav Hajek.