On the central limit theorem in $R^k$

An estimate of the rate of convergence in the central limit theorem in $R^k$ for i. i. d. summands in the spirit of [1] is obtained. The difference of the present estimate from that of [1] is in the dependence of the right side of it on the distribution of a summand. This dependence now ameliorates as the number of summands increases. In the one-dimensional case a similar result was obtained in [2].