Multidimensional Hungarian construction for vectors with almost Gaussian smooth distributions

A multidimensional version of the results of Koml\'os, Major and Tusn\'ady for sums of independent random vectors with finite exponential moments is obtained in the particular case where the summands have smooth distributions which are close to Gaussian ones. The bounds obtained reflect this closeness. Furthermore, the results provide sufficient conditions for the existence of i.i.d. vectors $X_1, X_2,\dots$ with given distributions and corresponding i.i.d. Gaussian vectors $Y_1, Y_2,\dots$ such that, for given small $\varepsilon$, $${\mathbf P}\Big\{{\limsup\limits_{n\to\infty} \frac1{\log n}\Bigl|\,\sum\limits_{j=1}^n X_j- \sum\limits_{j=1}^n Y_j\,\Bigr|}\le \varepsilon\Big\}=1.$$