On the Skitovich-Darmois theorem for \(a\)-adic solenoids (Q2870982)

The classical Skitovich-Darmois theorem says that if \(n\geq 2,\) \(x_i\), \(i=1, 2,\dots,n,\) are independent random variables, \(\alpha_i, \beta_i\) are nonzero constants, and the linear forms \(L_1=\alpha_1x_1+\dots+\alpha_nx_n\) and \(L_2=\beta_1x_1+\dots+\beta_nx_n\) are independent, then the random variables \(x_i\) are Gaussian [\textit{A. M. Kagan} et al., Characterization problems in mathematical statistics. Translated from Russian text by B. Ramachandran. New York etc.: John Wiley \& Sons (1973; Zbl 0271.62002)]. A weak analogue of the Skitovich-Darmois theorem fails for a compact connected ablian group even in the case when \(n=2\) [\textit{G. M. Feldman} and \textit{P. Graczyk}, J. Theor. Probab. 13, No. 3, 859--869 (2000; Zbl 0973.60011)].NEWLINENEWLINEIn this paper, the author shows that a weak analogue of the Skitovich-Darmois theorem holds for some compact connected Abelian groups if one considers three linear forms of three random variables. The author indeed constructs an \(a\)-adic solenoid \(\Sigma_a\) for which the independence of three linear forms of three independent random variables with values in \(\Sigma_a\) implies that at least one random variable has an idempotent distribution.