The Klebanov theorem for the group $\mathbb{R}\times \mathbb{Z}(2)$

TL;DR

This paper extends Klebanov's theorem to the group 1d7 2Z(2), characterizing Gaussianity of variables via distributional symmetries of linear forms with endomorphic coefficients.

Contribution

It provides a novel analogue of Klebanov's theorem for random variables valued in 1d7 2Z(2) with linear forms defined by topological endomorphisms.

Findings

Characterization of Gaussian variables in 1d7 2Z(2)

Extension of Klebanov's theorem to new group setting

Conditions for distributional symmetry implying Gaussianity

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 analogue of the Klebanov theorem in the case when random variables take values in the group $\mathbb{R}\times \mathbb{Z}(2)$ and the coefficients of the linear forms are topological endomorphisms of this group.