On the Skitovich-Darmois theorem for a-addic solenoids

Abstract : Let $X$ be a compact connected Abelian group. It is well-known that then there exist topological automorphisms $\alpha_j, \beta_j $ of $X$ and independent random variables $\xi_1$ and $\xi_2$ with values in $X$ and distributions $\mu_1, \mu_2$ such that the linear forms $L_1 = \alpha_1\xi_1 + \alpha_2\xi_2$ and $L_2 = \beta_1\xi_1 + \beta_2\xi_2$ are independent, but $\mu_1$ and $\mu_2$ are not represented as convolutions of Gaussian and idempotent distributions. To put this in other words in this case even a weak analogue of the Skitovich-Darmois theorem does not hold. We prove that there exists a compact connected Abelian group such that if we consider three linear forms of three independent random variables taking values in $X$ and the linear forms are independent, then at least one of the distributions is idempotent.
Document type :
Preprints, Working Papers, ...
Liste complète des métadonnées

Cited literature [6 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00735592
Contributor : Ivan Mazur <>
Submitted on : Wednesday, October 3, 2012 - 5:56:37 AM
Last modification on : Wednesday, December 19, 2018 - 2:08:04 PM
Document(s) archivé(s) le : Friday, January 4, 2013 - 3:55:38 AM

File

Mazur_Sol.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00735592, version 1

Collections

Citation

Ivan Mazur. On the Skitovich-Darmois theorem for a-addic solenoids. 2012. 〈hal-00735592〉

Share

Metrics

Record views

156

Files downloads

58