The weak convergence of step processes to a homogeneous Poisson process with independent increments

One obtains conditions for the convergence of the step processes _n(t) to a homogeneous Poisson process with independent increments, where, t_nm_n is a sequence of partitions of the segment [0, 1], while, ..., nm_n is a sequence of series of independent random variables, satisfying the condition of limiting uniform smallness.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 293–314, 1986.