Estimates for the rate of strong approximation in the multidimensional invariance principle

The aim of this paper is to derive simplest consequences of the author's result [17]. We obtain bounds for the rate of strong Gaussian approximation of sums of independent $\mathbf R^d$-valued random variables $\xi_j$ having finite moments of the form $\mathbf E\,H(\|\xi_j\|)$, where $H(x)$ is a monotone function growing not slower than $x^2$ and not faster than $e^{cx}$. A multidimensional version of the results of Sakhanenko [11] is obtained.