On the Square Root of Wishart Matrices: Exact Distributions and Asymptotic Gaussian Behavior
Random matrix theory has become a cornerstone in modern statistics and data science, providing fundamental tools for understanding high-dimensional covariance structures. Within this framework, the Wishart matrix plays a central role in multivariate analysis and related applications. This paper investigates both the exact and asymptotic distributions of the square root of a standard Wishart matrix. We first derive the exact distribution of the square root matrix. Then, by leveraging the Bartlett decomposition, we establish the joint asymptotic normality of the upper-triangular entries of the square root matrix. The resulting limiting distribution resembles that of a scaled Gaussian Wigner ensemble. Additionally, we quantify the rate of convergence using the 1-Wasserstein distance. To validate our theoretical findings, we conduct extensive Monte Carlo simulations, which demonstrate rapid convergence even with relatively low degrees of freedom. These results offer refined insights into the asymptotic behavior of random matrix functionals.