On the Skitovich-Darmois theorem for a-adic solenoids
Ivan Mazur

TL;DR
This paper extends the Skitovich-Darmois theorem to a-adic solenoids, showing that for three linear forms of three independent variables, independence implies at least one distribution is idempotent, unlike the classical case.
Contribution
It proves a new version of the Skitovich-Darmois theorem for a-adic solenoids involving three variables, where previously the theorem failed for two variables.
Findings
Independence of three linear forms implies at least one distribution is idempotent.
The classical Skitovich-Darmois theorem does not hold for two variables on these groups.
Characterization of all a-adic solenoids where this property holds.
Abstract
Let be a compact connected Abelian group. It is well-known that then there exist topological automorphisms of and independent random variables and with values in and distributions such that the linear forms and are independent, whereas and 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 , where is an -adic solenoid, then the independence of the linear forms implies that at least one of the distributions is idempotent. We describe all such solenoids.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
Topicsadvanced mathematical theories · Scientific Research and Philosophical Inquiry
