The Klebanov theorem for the group $\mathbb{R}\times \mathbb{Z}(2)$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912740896931840 |
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| author | Myronyuk, Margaryta |
| author_facet | Myronyuk, Margaryta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01689 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Klebanov theorem for the group $\mathbb{R}\times \mathbb{Z}(2)$ Myronyuk, Margaryta Probability Statistics Theory Primary 60B15, Secondary 62E10 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. |
| title | The Klebanov theorem for the group $\mathbb{R}\times \mathbb{Z}(2)$ |
| topic | Probability Statistics Theory Primary 60B15, Secondary 62E10 |
| url | https://arxiv.org/abs/2512.01689 |