Geodesic
Estimates of distances between sums of independent random elements in Banach spaces
Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 49-64

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Various metrics on the set of random elements in Banach space are treated. Estimates of closeness in these metrics of sums of independent random elements are obtained. Using these estimates we obtain the unimprovable rate of convergence in the central limit theorem for Banach space valued random elements in the Prochorov metric. 
@article{TVP_1984_29_1_a4,
     author = {V. Bentkus and A. Ra\v{c}kauskas},
     title = {Estimates of distances between sums of independent random elements in {Banach} spaces},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {49--64},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1984_29_1_a4/}
}
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V. Bentkus; A. Račkauskas. Estimates of distances between sums of independent random elements in Banach spaces. Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 49-64. http://geodesic.mathdoc.fr/item/TVP_1984_29_1_a4/