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Tokyo Probability Seminar
Seminar information archive ～05/22｜Next seminar｜Future seminars 05/23～
| Date, time & place | Monday 16:00 - 17:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Makiko Sasada, Shuta Nakajima (Keio Univ.), Masato Hoshino (Science Tokyo), Masahisa Ebina (Science Tokyo) |
2026/05/18
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Yoshihiro Gyotoku (The University of Tokyo)
Independence preservation property through web geometry
We are having teatime from 15:15 in the common room on the second floor. Please join us.
Yoshihiro Gyotoku (The University of Tokyo)
Independence preservation property through web geometry
[ Abstract ]
The subclass [2:2] of quadrirational Yang–Baxter maps on (0,∞)² contains three involutions H⁺_I, H⁺_II, and H^A_III. The central object of study is the independence-preserving property: given a map F with F(X,Y) = (U,V), one seeks all distributions of an independent pair (X,Y) for which (U,V) is again an independent pair. This property stands in direct analogy with classical characterisation theorems — the Kac–Bernstein and Lukacs theorems, and the Matsumoto–Yor property — in which the independence of a prescribed transformation characterises a specific family of distributions. A uniform derivation of the complete characterisation for all three maps is obtained via the theory of planar webs: a Jacobian identity common to all three maps reduces the problem to the determination of Abelian relations of a planar 4-web, whereupon Bol's bound and an explicit basis of three relations yield the full three-parameter families — the generalised beta-prime laws for H⁺_I, the Kummer laws for H⁺_II, and the generalised inverse Gaussian laws for H^A_III.
The subclass [2:2] of quadrirational Yang–Baxter maps on (0,∞)² contains three involutions H⁺_I, H⁺_II, and H^A_III. The central object of study is the independence-preserving property: given a map F with F(X,Y) = (U,V), one seeks all distributions of an independent pair (X,Y) for which (U,V) is again an independent pair. This property stands in direct analogy with classical characterisation theorems — the Kac–Bernstein and Lukacs theorems, and the Matsumoto–Yor property — in which the independence of a prescribed transformation characterises a specific family of distributions. A uniform derivation of the complete characterisation for all three maps is obtained via the theory of planar webs: a Jacobian identity common to all three maps reduces the problem to the determination of Abelian relations of a planar 4-web, whereupon Bol's bound and an explicit basis of three relations yield the full three-parameter families — the generalised beta-prime laws for H⁺_I, the Kummer laws for H⁺_II, and the generalised inverse Gaussian laws for H^A_III.
