Geodesic
Independent random variables on Abelian groups with independent sum and difference
Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 404-414

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Let $X$ be a second countable locally compact Abelian group. Let $\xi_1$, $\xi_2$ be independent random variables with values in the group $X$ and distributions $\mu_1$, $\mu_2$ such that the sum $\xi_1+\xi_2$ and the difference $\xi_1-\xi_2$ are independent. Assuming that the connected component of the zero of group $X$ contains a finite number of elements of order 2, we describe the possible distributions $\mu_k$. 
@article{TVP_2016_61_2_a11,
     author = {G. M. Feldman},
     title = {Independent random variables on {Abelian} groups with independent sum and difference},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {404--414},
     publisher = {mathdoc},
     volume = {61},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a11/}
}
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G. M. Feldman. Independent random variables on Abelian groups with independent sum and difference. Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 404-414. http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a11/