Geodesic
Central limit theorem for stationary sequences in the Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 337-339

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Let $\xi_i$ be a centered strong stationary sequence in the separable Hilbert space $H$. We say that $\xi_i$ satisfy CLT if $S_n=n^{-1/2}(\xi_1+\dots+\xi_n)$ converges weakly to a Gaussian variable $\eta$, $\mathbf P\{\eta\in H\}=1$. We study some conditions on a mixing of a sequence $\xi_i$ for this sequence to satisfy CLT. 
@article{TVP_1982_27_2_a11,
     author = {V. V. Mal'cev and E. I. Ostrovskiǐ},
     title = {Central limit theorem for stationary sequences in the {Hilbert} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {337--339},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a11/}
}
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V. V. Mal'cev; E. I. Ostrovskiǐ. Central limit theorem for stationary sequences in the Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 337-339. http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a11/