On a characterization of gaussian measures in a Hilbert space
Annales de l'I.H.P. Probabilités et statistiques, Tome 11 (1975) no. 4, p. 397-404 
@article{AIHPB_1975__11_4_397_0,
     author = {Kannan, D. and Kannappan, Pl.},
     title = {On a characterization of gaussian measures in a Hilbert space},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {11},
     number = {4},
     year = {1975},
     pages = {397-404},
     zbl = {0322.60005},
     mrnumber = {394787},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_1975__11_4_397_0}
}

Kannan, D.; Kannappan, Pl. On a characterization of gaussian measures in a Hilbert space. Annales de l'I.H.P. Probabilités et statistiques, Tome 11 (1975) no. 4, pp. 397-404. http://www.numdam.org/item/AIHPB_1975__11_4_397_0/

[1] L. Corwin, Generalized Gaussian measures and a « Functional equation », III. Advances in Math., vol. 6, 1971, p. 239-251. | MR 273295 | Zbl 0214.13601

[2] W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, John Wiley, New York, 1966. | MR 210154 | Zbl 0138.10207

[3] M. Fisz, Probability Theory and Mathematical Statistics. John Wiley, New York, 1963. | MR 164358 | Zbl 0123.34504

[4] I.I. Gikhman and A.V. Skorokhod, On the densities of probability measures in function spaces. Russian Math. Surveys, vol. 6, 1966, p. 83-156. | Zbl 0203.17101

[5] M. Kac, On a characterization of the normal distribution. Amer. J. Math., vol. 61, 1939, p. 726-728. | MR 371 | Zbl 0022.06102

[6] K.R. Parthasarathy, Probability Measures on Metric Spaces. Academic Press, New York, 1967. | MR 226684 | Zbl 0153.19101

[7] Yu.V. Prokhorov and M. Fish, A characterization of normal distributions in Hilbert space. Theory Prob. Appl., vol. 2, 1957, p. 468-469.