Geodesic
Quasi-probability Wigner densities and characterization of Gaussian distribution
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 1, Tome 228 (1996), pp. 142-153

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A new characterization of the Gaussian distribution by nonnegativity of a quazi-probability Wigner density was established by R. Hudson in 1974. Some sharpenings and extensions of this result are proposed, and connections with some other characterizations of Gaussian distribution are established. Linnik's “ridge principle” is essentially used. Bibl. 11 titles. 
@article{ZNSL_1996_228_a11,
     author = {A. A. Zinger},
     title = {Quasi-probability {Wigner} densities and characterization of {Gaussian} distribution},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {142--153},
     publisher = {mathdoc},
     volume = {228},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1996_228_a11/}
}
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A. A. Zinger. Quasi-probability Wigner densities and characterization of Gaussian distribution. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 1, Tome 228 (1996), pp. 142-153. http://geodesic.mathdoc.fr/item/ZNSL_1996_228_a11/