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

TL;DR

This work generalizes the Klebanov theorem from the real line to the nonconnected locally compact Abelian group X = $\mathbb{R}\times\mathbb{Z}(2)$, using linear forms with coefficients in End(X). By analyzing the joint distributions via characteristic functions and applying a finite-difference approach, the author identifies three distinct outcomes: (1) if the nondegeneracy condition a_i d_j − b_i c_j ∈ Aut(X) holds for all relevant indices, the involved ξi must be Gaussian on X; (2) if a'_i d'_j − b'_i c'_j ≠ 0 for all j and at least one a''_i or b''_i equals 1, then ξi fall into the Θ(X) class; (3) if a'_i d'_j − b'_i c'_j ≠ 0 for all j, then ξi belong to Λ(X). These results extend the Klebanov framework to a simple yet nontrivial LCA group and connect to group analogues of Darmois–Skitovich and Heyde theorems, illustrating how the connected/discrete structure influences Gaussianity characterizations on groups.

Abstract

L. Klebanov proved the following theorem. Let $ξ_1, \dots, ξ_n$ be independent random variables. Consider linear forms $L_1=a_1ξ_1+\cdots+a_nξ_n,$ $L_2=b_1ξ_1+\cdots+b_nξ_n,$ $L_3=c_1ξ_1+\cdots+c_nξ_n,$ $L_4=d_1ξ_1+\cdots+d_nξ_n,$ 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 $ξ_i$ 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.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Proposition 1