Geodesic
Estimates of the accuracy of normal approximation in a~Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 279-285

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Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables with values in a separable Hilbert space such that $\mathbf EX_j=0$, $\mathbf E|x_j|^{3+\delta}\infty$, $0\le\delta\le 1$. Estimates of the accuracy of normal approximation for $\mathbf P\{|n^{-1/2}(X_1+\dots+X_n)|$ are constructed. For $0\le\delta\le 1$ the order of approximation is $O(n^{-1_+\delta)/2})$, for $\delta=1$ the order is $O(n^{-1+\varepsilon})$, $\varepsilon>0$. 
@article{TVP_1982_27_2_a6,
     author = {B. A. Zalesskiǐ},
     title = {Estimates of the accuracy of normal approximation in {a~Hilbert} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {279--285},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a6/}
}
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B. A. Zalesskiǐ. Estimates of the accuracy of normal approximation in a~Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 279-285. http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a6/