Geodesic
On the Skitovich--Darmois Theorem for $\mathbf{a}$-Adic Solenoids
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013), pp. 582-593

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By the Skitovich–Darmois theorem, the Gaussian distribution on the real line is characterized by the independence of two linear forms of $n$ independent random variables. The theorem is known to fail for a compact connected Abelian group even in the case when $n=2$. In the paper, it is proved that a weak analogue of the Skitovich–Darmois theorem holds for some $\mathbf{a}$-adic solenoids if we consider three independent linear forms of three random variables. 
@article{JMAG_2013_9_a6,
     author = {I. P. Mazur},
     title = {On the {Skitovich--Darmois} {Theorem} for $\mathbf{a}${-Adic} {Solenoids}},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {582--593},
     publisher = {mathdoc},
     volume = {9},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2013_9_a6/}
}
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I. P. Mazur. On the Skitovich--Darmois Theorem for $\mathbf{a}$-Adic Solenoids. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013), pp. 582-593. http://geodesic.mathdoc.fr/item/JMAG_2013_9_a6/