04.03.2026
Yoshihiro Gyotoku, Graduate School of Mathematical Sciences, The University of Tokyo
"Independence Preservation and Planar Webs: the Quadrirational Yang-Baxter Case"
Abstract: Independence-preserving transformations are central in classical characterisations such as Lukacs' theorem and the Matsumoto-Yor property. In this talk, the emphasis is on a more recent source of such phenomena: quadrirational Yang-Baxter maps (of Adler-Bobenko-Suris type \([2:2]\)), viewed as real-analytic involutions on \((0, \infty)^2\).
Given a diffeomorphism \(F\) on a product domain, the requirement that independent inputs \((X,Y)\) yield independent outputs \((U,V)=F(X,Y)\) for some product law is equivalent to a separated functional equation for the logarithms of the densities. The key point is that with this point of view, the characterisation of all independence-preserving product measures reduces to describing the Abelian relations of the associated planar \(4\)-web given by the foliations \((x, y, u, v)\). Using the Bol bound for planar \(4\)-webs, one obtains that for each of the three quadrirational involutions \(H_I^+, H_{II}^+\) and \(H_{III}^A\), the space of Abelian relations has dimension \(3\), and explicit bases can be written down. This yields a complete three-parameter family of independence-preserving product distributions, which is beta-prime, Kummer and GIG-type families.
Yoshihiro Gyotoku, Graduate School of Mathematical Sciences, The University of Tokyo
"Independence Preservation and Planar Webs: the Quadrirational Yang-Baxter Case"
Abstract: Independence-preserving transformations are central in classical characterisations such as Lukacs' theorem and the Matsumoto-Yor property. In this talk, the emphasis is on a more recent source of such phenomena: quadrirational Yang-Baxter maps (of Adler-Bobenko-Suris type \([2:2]\)), viewed as real-analytic involutions on \((0, \infty)^2\).
Given a diffeomorphism \(F\) on a product domain, the requirement that independent inputs \((X,Y)\) yield independent outputs \((U,V)=F(X,Y)\) for some product law is equivalent to a separated functional equation for the logarithms of the densities. The key point is that with this point of view, the characterisation of all independence-preserving product measures reduces to describing the Abelian relations of the associated planar \(4\)-web given by the foliations \((x, y, u, v)\). Using the Bol bound for planar \(4\)-webs, one obtains that for each of the three quadrirational involutions \(H_I^+, H_{II}^+\) and \(H_{III}^A\), the space of Abelian relations has dimension \(3\), and explicit bases can be written down. This yields a complete three-parameter family of independence-preserving product distributions, which is beta-prime, Kummer and GIG-type families.
Everyone is cordially invited!
B. Kołodziejek, W. Matysiak, K. Szpojankowski, J. Wesołowski