Geodesic
Bounds for the Rate of Strong Approximation in the Multidimensional Invariance Principle
Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 1, pp. 100-123

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The goal of this paper is to derive consequences of the result of Zaitsev [Theory Probab. Appl., 45 (2001), pp. 624–642; 46 (2002), pp. 490–514; 676–698]. We establish bounds for the rate of strong Gaussian approximation of sums of independent $\mathbf{R}^d$-valued random vectors $\xi_j$ having finite moments $\mathbf{E}\|\xi_j\|^\gamma$, $\gamma\ge 2$. A multidimensional version of the results of Sakhanenko [Trudy Inst. Mat., 5 (1985), pp. 27–44 (in Russian)] is obtained.
Keywords: multidimensional invariance principle, strong approximation, sums of independent random vectors. 
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     title = {Bounds for the {Rate} of {Strong} {Approximation} in the {Multidimensional} {Invariance} {Principle}},
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F. Götze; A. Yu. Zaitsev. Bounds for the Rate of Strong Approximation in the Multidimensional Invariance Principle. Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 1, pp. 100-123. http://geodesic.mathdoc.fr/item/TVP_2008_53_1_a5/