Estimates for the rate of strong Gaussian approximation for the sums of i.i.d. multidimensional random vectors

The aim of this paper is to derive new optimal bounds for the rate of strong Gaussian approximation of sums of i.i.d. $\mathbb R^d$-valued random variables $\xi_j$ having finite moments of the form $\mathbb{E}\,H(\|\xi_j\|)$, where $H(x)$ is a monotone function growing not slower than $x^2$ and not faster than $e^{cx}$. We obtain some generalization and improvements of the results of U. Einmahl (1989).