The Kagan characterization theorem on Banach spaces

TL;DR

This paper extends Kagan's characterization theorem, originally for finite-dimensional spaces, to certain Banach spaces, identifying conditions under which Gaussian distributions are characterized by linear forms.

Contribution

It generalizes Kagan's theorem from finite-dimensional Euclidean spaces to specific Banach spaces, establishing when Gaussianity can be characterized similarly.

Findings

Identifies Banach spaces where Kagan's theorem holds

Provides conditions for Gaussian characterization in Banach spaces

Extends classical finite-dimensional results to infinite-dimensional settings

Abstract

A. Kagan introduced classes of distributions $\mathcal{D}_{m,k}$ in $m$-dimensional space $\mathbb{R}^m$. He proved that if the joint distribution of $m$ linear forms of $n$ independent random variables belong to the class $\mathcal{D}_{m,m-1}$ then the random variables are Gaussian. If $m=2$ then the Kagan theorem implies the well-known Darmois-Skitovich theorem, where the Gaussian distribution is characterized by the independence of two linear forms of $n$ independent random variables. In the paper we describe Banach spaces where the analogue of the Kagan theorem is valid.