Geodesic
Two characteristic properties of normal distributions in the class~$NH$
Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 139-146.

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In this note are deduced two characterizations of normal distributions in the class $NH$ of probability distributions on a plane whose densities admit diagonal expansions in series of Hermite polynomials and the marginal distributions are standardized and normal. 
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     title = {Two characteristic properties of normal distributions in the class~$NH$},
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V. A. Romanovich; N. V. Lobaskina. Two characteristic properties of normal distributions in the class~$NH$. Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 139-146. http://geodesic.mathdoc.fr/item/MZM_1976_20_1_a14/