Estimates for the strong approximation in multidimensional central limit theorem

In a recent paper the author obtained optimal bounds for the strong Gaussian approximation of sums of independent $\R^d$-valued random vectors with finite exponential moments. The results may be considered as generalizations of well-known results of Koml\'os--Major--Tusn\'ady and Sakhanenko. The dependence of constants on the dimension $d$ and on distributions of summands is given explicitly. Some related problems are discussed.