Jones-Balakrishnan Property for Matrix Variate Beta Distributions

Let X and Y be independent m × m symmetric positive definite random matrices. Assume that X follows a matrix variate beta distribution with parameters a and b and that Y has a matrix variate beta distribution with parameters a + b and c. Define R=Im−Y+Y1/2XY1/2−1/2Y1/2XY1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {R}= \left (\boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{-1/2} \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}$\end{document}Im−Y+Y1/2XY1/2−1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \left (\boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{-1/2} $\end{document} and S=Im−Y+Y1/2XY1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {S}= \boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}$\end{document}, where Im is an identity matrix and A1/2 is the unique symmetric positive definite square root of A. In this paper, we have shown that random matrices R and S are independent and follow matrix variate beta distributions generalizing an independence property established by Jones and Balakrishnan (Statistics and Probability Letters, 170 (2021), article id 109011) in the univariate case.