Geodesic
Some estimates of the accuracy of approximation in the multidimensional central limit theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 621-624

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The paper contains some estimates of the remainder term in the central limit theorem for the case of general Banach space and for its particular case $R_m$ ($m\le 5$). In the case of general Banach space the special metrics $\lambda_k$ are used, which are defined by means of characteristic functions. In the case of $R_m$ ($m\le 5$) we use the well-known Levy–Prohorov metrics. 
@article{TVP_1979_24_3_a18,
     author = {I. S. \v{S}iganov},
     title = {Some estimates of the accuracy of approximation in the multidimensional central limit theorem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {621--624},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a18/}
}
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I. S. Šiganov. Some estimates of the accuracy of approximation in the multidimensional central limit theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 621-624. http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a18/