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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 2×2 random matrices including beta matrices is also obtained. The main technical challenge was a problem involving the solution of a related functional equation.
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Bobecka et al. (2008) studied this question.
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