The Independence-Preserving Property and Planar Web Geometry

Abstract

This paper introduces planar web geometry into the study of characterization problems for probability laws arising from independence preservation. Let $F(x,y) = (u, v)$ be a real-analytic diffeomorphism defined on an open domain in $\mathbb{R}^2$. Let $X$ and $Y$ be independent random variables and set $(U, V) = F(X, Y)$. The map $F$ is said to preserve the independence if $U$ and $V$ are independent. This condition is highly restrictive and can therefore be used to characterize the laws for which it holds. This framework provides unified proofs of characterization theorems, including classical results such as the Kac--Bernstein and Lukacs theorems and the Matsumoto--Yor property, as well as recent results for the quadrirational Yang--Baxter maps $H_I^+$, $H_{II}^+$, and $H_{III}^A$. For measures with positive continuous densities, the map preserves independence if and only if their logarithmic densities solve the corresponding inhomogeneous Abelian functional equation. The solution space is identified with an affine space modeled on the space of Abelian relations, whose dimension is at most three according to Bol's rank bound. The same method also produces further examples, including a class generalizing transformations studied by Koudou and Vallois; every map in this class admits at least a two-parameter family of preserved product measures. The independence-preserving property of a map is preserved by coordinatewise one-variable reparametrizations, which also preserve the associated web. Thus the web associated with a map is a natural invariant under the corresponding equivalence relation of independence-preserving maps. Together with the local algebraizability of nonsingular planar $4$-webs of maximal rank, this correspondence suggests a route toward a complete classification of maps preserving the independence of a three-parameter family of measures.