Geodesic
Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 4, pp. 741-758 
V. M. Zolotarev. Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 4, pp. 741-758. http://geodesic.mathdoc.fr/item/TVP_1976_21_4_a4/
@article{TVP_1976_21_4_a4,
     author = {V. M. Zolotarev},
     title = {Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {741--758},
     year = {1976},
     volume = {21},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_4_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The problem under consideration is to estimate the distance, with respect to a chosen metric $\mu$, between two linear combinations $\displaystyle X=\sum_jc_jX_j$ and $\displaystyle Y=\sum_jc_jY_j$ of independent random variables with values in a Banach space $U$. General results of this paper enable, in particular, to effectively estimate the accuracy of approximation of the distributions of normalized sums of independent random $U$-valued variables by a normal law. When choosing $\mu$ in an appropriate way, one obtains estimates quite analogous to those known in the simplest case $U=R^1$.