On the Skitovich-Darmois theorem for a-adic solenoids

Abstract

Let $X$ be a compact connected Abelian group. It is well-known that then there exist topological automorphisms $α_j, β_j $ of $X$ and independent random variables $ξ_1$ and $ξ_2$ with values in $X$ and distributions $μ_1, μ_2$ such that the linear forms $L_1 = α_1ξ_1 + α_2ξ_2$ and $L_2 = β_1ξ_1 + β_2ξ_2$ are independent, whereas $μ_1$ and $μ_2$ are not represented as convolutions of Gaussian and idempotent distributions. This means that the Skitovich--Darmois theorem fails for such groups. We prove that if we consider three linear forms of three independent random variables taking values in $X$, where $X$ is an ${\boldsymbol a}$-adic solenoid, then the independence of the linear forms implies that at least one of the distributions is idempotent. We describe all such solenoids.