Geodesic
The rate of Gaussian strong approximation for the sums of i.i.d. multidimensional random vectors
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 14–2, Tome 364 (2009), pp. 148-165

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The aim of this paper is to derive new optimal bounds for the rate of strong Gaussian approximation of sums of i.i.d. $\mathbf R^d$-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 $x^{2+\delta}$ and not faster than $e^{cx}$. We obtain some generalizations of the results of U. Einmahl (1989). Bibl. – 44 titles. 
@article{ZNSL_2009_364_a6,
     author = {A. Yu. Zaitsev},
     title = {The rate of {Gaussian} strong approximation for the sums of i.i.d. multidimensional random vectors},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {148--165},
     publisher = {mathdoc},
     volume = {364},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_364_a6/}
}
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A. Yu. Zaitsev. The rate of Gaussian strong approximation for the sums of i.i.d. multidimensional random vectors. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 14–2, Tome 364 (2009), pp. 148-165. http://geodesic.mathdoc.fr/item/ZNSL_2009_364_a6/