# On a generalisation of the Skitovich--Darmois theorem for several linear forms on Abelian groups

## TL;DR

This paper extends Kagan's theorem, which characterizes Gaussian distributions via linear forms, to a broad class of locally compact Abelian groups, generalizing the Skitovich--Darmois theorem.

## Contribution

It generalizes Kagan's theorem to various Abelian groups, broadening the scope of distribution characterization via linear forms.

## Key findings

- Kagan's theorem holds on a wide class of Abelian groups
- The characterization of Gaussian distributions extends beyond real numbers
- The results unify and generalize classical distribution characterization theorems

## Abstract

A.M. Kagan introduced a class of distributions $\mathcal{D}_{m, k}$ in $\mathbb{R}^m$ and proved that if the joint distribution of $m$ linear forms of $n$ independent random variables belongs to the class $\mathcal{D}_{m, m-1}$, then the random variables are Gaussian. A.M. Kagan's theorem implies, in particular, the well-known Skitovich--Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of $n$ independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan's theorem is valid.