On the Skitovich-Darmois theorem for compact Abelian groups
The classical Skitovich-Darmois theorem characterizes Gaussian distribution by the independence of linear statistics. The authors consider the (compact and Abelian) group analogue of the Skitovich-Darmois theorem and a complete answer is given in the cases of totally disconnected compact groups and connected compact groups. (1) Let \(\psi _1\) and \(\psi _2\) be \(X\)-valued independent random variables where \(X\) is a separable totally disconnected compact Abelian group. Then the independence of \(\alpha _1 (\psi _1)+\alpha _2 (\psi _2)\) and \(\beta _1(\psi _1) +\beta _2 (\psi _2)\) where \(\alpha _i, \beta _i \in \Aut(X)\) implies that the distributions of \(\psi _1\) and \(\psi _2\) are idempotent if and only if for each prime \(p\), \(X/X^{(p)}\) is finite. It may be remarked that the set of Gaussian distributions on a totally disconnected group coincides with the set of degenerate distributions. (2) For a separable compact connected Abelian group \(X\), there are \(X\)-valued independent random variables \(\psi _1\) and \(\psi _2\) and automorphisms \(\alpha _i\) and \(\beta _i\) of \(X\) such that \(\alpha _1 (\psi _1) + \alpha _2 (\psi _2)\) and \(\beta _1 (\psi _1) +\beta _2 (\psi _2)\) are independent but the distribution of \(\psi _1\) or \(\psi _2\) is not Gaussian (up to shifts by an idempotent).
characterization of probability distribution
scientific article; zbMATH DE number 1552713