Certain class of nonuniform estimates in multidimensional limit theorems

Quantities of the form | F(X) — G(X) | are estimated, where F and G are the convolutions of certain k-dimensional probability distributions, while X is a convex polyhedron in R^k. Estimates of the form | F(X) — G(X) | c(k)(F, G, X) are proved, differing from the known ones by the presence of the factor (F, G, X) in the right-hand side, which may turn out to be small if the polyhedron X is small in a definite sense.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 92–105, 1990.