On the Li--Zheng theorem
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917642296623104 |
|---|---|
| author | Feldman, Gennadiy |
| author_facet | Feldman, Gennadiy |
| contents | By the well-known I.Kotlarski lemma, if $ξ_1$, $ξ_2$, and $ξ_3$ are independent real-valued random variables with nonvanishing characteristic functions, $L_1=ξ_1-ξ_3$ and $L_2=ξ_2-ξ_3$, then the distribution of the random vector $(L_1, L_2)$ determines the distributions of the random variables $ξ_j$ up to shift. Siran Li and Xunjie Zheng generalized this result for the linear forms $L_1=ξ_1+a_2ξ_2+a_3ξ_3$ and $L_2=b_2ξ_2+b_3ξ_3+ξ_4$ assuming that all $ξ_j$ have first and second moments, $ξ_2$ and $ξ_3$ are identically distributed, and $a_j$, $b_j$ satisfy some conditions. In the article, we give a simpler proof of this theorem. In doing so, we also prove that the condition of existence of moments can be omitted. Moreover, we prove an analogue of the Li--Zheng theorem for independent random variables with values in the field of $p$-adic numbers, in the field of integers modulo $p$, where $p\ne 2$, and in the discrete field of rational numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Li--Zheng theorem Feldman, Gennadiy Probability 39B52, 39A60, 60E05 By the well-known I.Kotlarski lemma, if $ξ_1$, $ξ_2$, and $ξ_3$ are independent real-valued random variables with nonvanishing characteristic functions, $L_1=ξ_1-ξ_3$ and $L_2=ξ_2-ξ_3$, then the distribution of the random vector $(L_1, L_2)$ determines the distributions of the random variables $ξ_j$ up to shift. Siran Li and Xunjie Zheng generalized this result for the linear forms $L_1=ξ_1+a_2ξ_2+a_3ξ_3$ and $L_2=b_2ξ_2+b_3ξ_3+ξ_4$ assuming that all $ξ_j$ have first and second moments, $ξ_2$ and $ξ_3$ are identically distributed, and $a_j$, $b_j$ satisfy some conditions. In the article, we give a simpler proof of this theorem. In doing so, we also prove that the condition of existence of moments can be omitted. Moreover, we prove an analogue of the Li--Zheng theorem for independent random variables with values in the field of $p$-adic numbers, in the field of integers modulo $p$, where $p\ne 2$, and in the discrete field of rational numbers. |
| title | On the Li--Zheng theorem |
| topic | Probability 39B52, 39A60, 60E05 |
| url | https://arxiv.org/abs/2404.10916 |