Geodesic
A Remark on Cramer’s Theorem on the Decomposition of the Normal Law
Teoriâ veroâtnostej i ee primeneniâ, Tome 1 (1956) no. 4, pp. 479-480

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H. Cramer's theorem is shown to be an equivalent of a particular case of Skitovich–Darmois' theorem on the independence of linear statistics. This means that each one of these theorems can be deduced from the other by a short elementary argument. 
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Yu. V. Linnik. A Remark on Cramer’s Theorem on the Decomposition of the Normal Law. Teoriâ veroâtnostej i ee primeneniâ, Tome 1 (1956) no. 4, pp. 479-480. http://geodesic.mathdoc.fr/item/TVP_1956_1_4_a4/