Independence preserving property of Kummer laws
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910303361433600 |
|---|---|
| author | Koudou, Efoevi Angelo Wesołowski, Jacek |
| author_facet | Koudou, Efoevi Angelo Wesołowski, Jacek |
| contents | We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $α,β\ge 0$, $α\neq β$, $$ ψ_{α,β}(x,y)=\left(y\,\tfrac{1+β(x+y)}{1+αx+βy},\;x\,\tfrac{1+α(x+y)}{1+αx+βy}\right), $$ then the random variables $U$ and $V$ defined by $(U,V)=ψ_{α,β}(X,Y)$ are independent if and only if $X$ and $Y$ follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by $ψ_{α,β}$ in the scheme introduced by Croydon and Sasada in \cite{CS2020} is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of $$ U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$ which corresponds to the case of $ψ_{1,0}$.
We also show that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03150 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Independence preserving property of Kummer laws Koudou, Efoevi Angelo Wesołowski, Jacek Probability Statistics Theory We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $α,β\ge 0$, $α\neq β$, $$ ψ_{α,β}(x,y)=\left(y\,\tfrac{1+β(x+y)}{1+αx+βy},\;x\,\tfrac{1+α(x+y)}{1+αx+βy}\right), $$ then the random variables $U$ and $V$ defined by $(U,V)=ψ_{α,β}(X,Y)$ are independent if and only if $X$ and $Y$ follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by $ψ_{α,β}$ in the scheme introduced by Croydon and Sasada in \cite{CS2020} is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of $$ U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$ which corresponds to the case of $ψ_{1,0}$. We also show that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models. |
| title | Independence preserving property of Kummer laws |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2212.03150 |