Identically distributed random vectors on locally compact Abelian groups

TL;DR

This paper extends Klebanov's theorem, which characterizes Gaussian variables via linear forms, to the setting of locally compact Abelian groups with integer coefficients, broadening the scope of the original result.

Contribution

It provides an analog of Klebanov's theorem for random variables on locally compact Abelian groups with integer coefficients in the linear forms.

Findings

Identifies conditions under which variables are Gaussian in the group setting.

Generalizes classical results to a broader algebraic structure.

Establishes new criteria for distributional equivalence in groups.

Abstract

L. Klebanov proved the following theorem. Let $\xi_1, \dots, \xi_n$ be independent random variables. Consider linear forms $L_1=a_1\xi_1+\cdots+a_n\xi_n,$ $L_2=b_1\xi_1+\cdots+b_n\xi_n,$ $L_3=c_1\xi_1+\cdots+c_n\xi_n,$ $L_4=d_1\xi_1+\cdots+d_n\xi_n,$ where the coefficients $a_j, b_j, c_j, d_j$ are real numbers. If the random vectors $(L_1,L_2)$ and $(L_3,L_4)$ are identically distributed, then all $\xi_i$ for which $a_id_j-b_ic_j\neq 0$ for all $j=\overline{1,n}$ are Gaussian random variables. The present article is devoted to an analog of the Klebanov theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.