Let r be the number of elements of a sample taken from a population uniformly distributed in [0,1]. Let ≥ 0 be a number such that λ=n ≤ 1. Subdivide an interval of length 1 - λ into n parts by n - 1 independently and uniformly distributed points. Denote   the lengths of these subintervals. Using the notations   the generalizations of the theorems of SMIRNOV are expressed by (4.14), (4.15), where G_m(t, μ) is defined similarly to F_n(t, λ) and these two stochastic processes are aupposec! to be independent. These theorems were already published in [7] the proofs are given here. Applications to inventory problems are also mentioned.