On a characterization theorem in the space $\mathbb{R}^n$

TL;DR

This paper extends Heyde's characterization theorem from the real line to higher-dimensional spaces, identifying the class of distributions as convolutions of Gaussian and subspace-supported distributions.

Contribution

It generalizes Heyde's theorem to $ extbf{R}^n$, describing the structure of distributions through linear forms of independent vectors.

Findings

Characterization of distributions as convolutions of Gaussian and subspace-supported distributions.

Extension of Heyde's theorem to multidimensional spaces.

Identification of the class of distributions based on linear form symmetry.

Abstract

By Heyde's theorem, the class of Gaussian distributions on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We prove an analogue of this theorem for two independent random vectors taking values in the space $\mathbb{R}^n$. The obtained class of distributions consists of convolutions of Gaussian distributions and a distribution supported in a subspace, which is determined by coefficients of the linear forms.