Geodesic
On the characterization of multidimensional normal law by the independence of linear statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 381-385

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Let $\{X_j\}$ be a sequence of independent random vectors in $R^k$ and $\{A_j,B_j\}$ be a sequence of pairs of nonsingular real $(k\times k)$-matrices. It is shown that every $X_j$ has $k$-dimensional normal distribution if linear statistics (1) converge with probability 1 to independent random vectors and the condition (2) is satisfied. 
@article{TVP_1979_24_2_a10,
     author = {A. A. Zinger},
     title = {On the characterization of multidimensional normal law by the independence of linear statistics},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {381--385},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a10/}
}
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A. A. Zinger. On the characterization of multidimensional normal law by the independence of linear statistics. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 381-385. http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a10/