On the Skitovich-Darmois theorem for a-addic solenoids

Ivan Mazur (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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.

Domaines: Probabilités [math.PR]

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