Kshirsagar--Tan independence property of beta matrices and related characterizations

A new independence property of univariate beta distributions, related to the results of Kshirsagar and Tan for beta matrices, is presented. Conversely, a characterization of univariate beta laws through this independence property is proved. A related characterization of a family of $2\times2$ random matrices including beta matrices is also obtained. The main technical challenge was a problem involving the solution of a related functional equation.