An estimate in the multidimensional central limit theorem is obtained, which is used together with the Strassen-Dudley theorem to prove a strong approximation theorem for partial sums of independent, identically distributed $d$-dimensional random vectors. This theorem implies immediately multi-dimensional versions of the strong invariance principles of Strassen and Major as well as a new $d$-dimensional strong invariance principle which improves the known results for the 1-dimensional case. In particular, we are able to weaken the assumption in Major's strong invariance principle. At the same time, it is shown that the assumptions of our theorem are nearly necessary.
"Strong Invariance Principles for Partial Sums of Independent Random Vectors." Ann. Probab. 15 (4) 1419 - 1440, October, 1987. https://doi.org/10.1214/aop/1176991985