Probability laws associated to the independence preserving quadrirational Yang-Baxter maps -- the ultimate case

A map $F\colon\mathcal X\times\mathcal Y\to \mathcal U\times \mathcal V$ is said to be independence preserving (IP) if there exists a pair of independent random variables $(X,Y)$ valued in $\mathcal X\times\mathcal Y$ such that the two coordinates of $(U,V)=F(X,Y)$ are also independent. Recently, Sasada and Uozumi (2024) observed that a hierarchy of quadrirational Yang-Baxter maps gives rise to independence preserving transformations, and identified corresponding families of probability distributions. In view of the limiting properties of these IP maps, the newly defined generalized second kind beta ($\mathrm{GB}_{II}$) model stands at the top of the hierarchy: for independent random variables $X$ and $Y$ following a $\mathrm{GB}_{II}$ distribution, Sasada and Uozumi (2024) showed that when a special quadrirational Yang-Baxter map $F^{(\alpha,\beta)}$, parameterized by $(\alpha,\beta)\in(0,\infty)^2$, is applied to the pair $(X,Y)$, it produces another pair $(U,V)$ of independent $\mathrm{GB}_{II}$-distributed random variables. The aim of this paper is to show that the IP property of $F^{(\alpha,\beta)}$ uniquely identifies distributions of $X,Y,U$ and $V$ as belonging to the $\mathrm{GB}_{II}$ family. To this end, we introduce specially designed Laplace-type transforms. First, we carefully explain the connection between the results from Sasada and Uozumi (2024) and Koudou and Vallois (2012). Next, we focus on the characterization of the second kind beta and the generalized second kind beta distributions through the IP map $F^{(\alpha,\infty)}$. Finally, extending considerably the methodology developed for the case $(\alpha,\infty)$, we prove the characterization of $\mathrm{GB}_{II}$ distributions in the case $(\alpha,\beta)\in(0,\infty)^2$ with $\alpha\neq\beta$, which implies uniqueness in the ultimate missing case of the quadrirational Yang-Baxter hierarchy of IP models.