Geodesic
Characterization of a~class of infinitely divisible distributions in Hilbert space
Matematičeskie zametki, Tome 16 (1974) no. 5, pp. 777-782.

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It is established that the spectral measure of an infinitely divisible distribution $F$ in a Hilbert space $H$ is concentrated in a sphere of finite radius if and only if the integral $\int_H\exp(\alpha\|x\|\ln(\|x\|+1))\,dF$ is finite for some number $\alpha>0$. If this integral is finite for any $\alpha>0$ then the infinitely divisible distribution $F$ is normal (maybe, degenerate). 
@article{MZM_1974_16_5_a11,
     author = {V. M. Kruglov},
     title = {Characterization of a~class of infinitely divisible distributions in {Hilbert} space},
     journal = {Matemati\v{c}eskie zametki},
     pages = {777--782},
     publisher = {mathdoc},
     volume = {16},
     number = {5},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_5_a11/}
}
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V. M. Kruglov. Characterization of a~class of infinitely divisible distributions in Hilbert space. Matematičeskie zametki, Tome 16 (1974) no. 5, pp. 777-782. http://geodesic.mathdoc.fr/item/MZM_1974_16_5_a11/