Estimates for the Levy-Prokhorov distance in terms of characteristic functions and some of their applications

One gives estimates of the Levy-Prokhorov distance between distributions in R^k in terms of integrals of the differences of the characteristic functions and of their derivatives, similar to Esseen's inequality.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 108–127, 1982.The author is grateful to I. A. Ibragimov for his interest in this paper.