Geodesic
The central limit theorem and the strong law of large numbers in $l_p\{X\}$-spaces, $1\le p+\infty$
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 4, pp. 802-812

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The central limit theorem is proved for independent identically distributed random elements having strong second order moments with values in a Banach space with a Shauder basis. It is shown that, if $X$ is a $G$-space, then $l_p\{X\}$, $2\le p\infty$, is a space of the same type. The central limit theorem is also proved for the case when $1\le p\le 2$ and $X$ is a $G$-space and for $l_p\{l_s\}$-spaces where $1\le p,s\le 2$. The strong law of large numbers in these spaces is studied. 
@article{TVP_1976_21_4_a8,
     author = {V. V. Kvara\v{c}heliya and Nguen Zuy Tien},
     title = {The central limit theorem and the strong law of large numbers in $l_p\{X\}$-spaces, $1\le p<+\infty$},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {802--812},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_4_a8/}
}
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V. V. Kvaračheliya; Nguen Zuy Tien. The central limit theorem and the strong law of large numbers in $l_p\{X\}$-spaces, $1\le p<+\infty$. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 4, pp. 802-812. http://geodesic.mathdoc.fr/item/TVP_1976_21_4_a8/