Geodesic
Estimates for the rate of strong approximation in the multidimensional invariance principle
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 10, Tome 339 (2006), pp. 37-53

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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. 
@article{ZNSL_2006_339_a2,
     author = {A. Yu. Zaitsev},
     title = {Estimates for the rate of strong approximation in the multidimensional invariance principle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {37--53},
     publisher = {mathdoc},
     volume = {339},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_339_a2/}
}
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A. Yu. Zaitsev. Estimates for the rate of strong approximation in the multidimensional invariance principle. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 10, Tome 339 (2006), pp. 37-53. http://geodesic.mathdoc.fr/item/ZNSL_2006_339_a2/