Geodesic
Optimal estimates for the rate of strong Gaussian approximation in the infinite dimensional invariance principle
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 93-101

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Estimates for the rate of strong Gaussian approximation in the invariance principle in the Hilbert space for sums of i.i.d. random vectors are derived. It is shown that they are optimal with respect to the order if the sequence of eigenvalues of the covariance operator of summands decreases slowly. 
@article{ZNSL_2011_396_a5,
     author = {A. Yu. Zaitsev},
     title = {Optimal estimates for the rate of strong {Gaussian} approximation in the infinite dimensional invariance principle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {93--101},
     publisher = {mathdoc},
     volume = {396},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a5/}
}
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A. Yu. Zaitsev. Optimal estimates for the rate of strong Gaussian approximation in the infinite dimensional invariance principle. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 93-101. http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a5/