On the Skitovich–Darmois Theorem for $\mathbf{a}$-Adic Solenoids

I. P. Mazur

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkiv 61103, Ukraine

Abstract: 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.

Key words and phrases: Skitovich–Darmois theorem, functional equation, $\mathbf{a}$-adic solenoid.

MSC: 39B52, 62E10, 60B15

Received: 10.07.2013

Language: English

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