Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Square Root of Wishart Matrices: Exact Distributions and Asymptotic Gaussian Behavior

Fengcheng Liu

TL;DR

This paper studies the symmetric square root $V=W^{1/2}$ of the standard Wishart matrix $W_p(m)$. It derives the exact density of $V$ via the matrix-square-root Jacobian and spectral decomposition, including explicit forms for the general case and the standard Wishart case $\Sigma=I_p$. It then shows that, as $m\to\infty$ with fixed $p$, the upper-triangular entries of $V$ become asymptotically normal and that $W_p(m)^{1/2}-\sqrt{m}I_p \xrightarrow{d} \tfrac{1}{2}G$, where $G$ is a Gaussian Wigner matrix; this is established using Bartlett decomposition and Slutsky's lemma. A key contribution is a concrete 1-Wasserstein convergence-rate bound of order $O(p^{5/2}/\sqrt{m})$ for the upper-triangular entries, together with supporting Monte Carlo simulations showing rapid convergence even for modest $m$. The results deepen the understanding of high-dimensional matrix functionals and open avenues for analyzing other transformations of Wishart matrices.

Abstract

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.

On the Square Root of Wishart Matrices: Exact Distributions and Asymptotic Gaussian Behavior

TL;DR

This paper studies the symmetric square root of the standard Wishart matrix . It derives the exact density of via the matrix-square-root Jacobian and spectral decomposition, including explicit forms for the general case and the standard Wishart case . It then shows that, as with fixed , the upper-triangular entries of become asymptotically normal and that , where is a Gaussian Wigner matrix; this is established using Bartlett decomposition and Slutsky's lemma. A key contribution is a concrete 1-Wasserstein convergence-rate bound of order for the upper-triangular entries, together with supporting Monte Carlo simulations showing rapid convergence even for modest . The results deepen the understanding of high-dimensional matrix functionals and open avenues for analyzing other transformations of Wishart matrices.

Abstract

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.

Paper Structure

This paper contains 6 sections, 12 theorems, 86 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Exact distribution of $(W_p(m))^{\frac{1}{2}}$
  3. Asymptotic distribution of $(W_p(m))^{\frac{1}{2}}$
  4. Convergence rate of $(W_p(m))^{\frac{1}{2}}$
  5. Numerical experiments
  6. Conclusion

Key Result

Proposition 1

Suppose that $W\sim W_p(m,\Sigma)$. Then, $W$ has the following density function where $\Gamma_p(a) = \pi^{p(p-1)/4} \prod_{i=1}^p\Gamma\left(a-\frac{i-1}{2} \right)$ denotes the multivariate Gamma function for $a>(p-1)/2$.

Figures (2)

  • Figure 1: Distribution and normal Q-Q comparison for $V_{11}-\sqrt{m-\frac{p}{4}}$ and $V_{12}$ with $(m,p)=(5,2)$
  • Figure 2: Distribution and normal Q-Q comparison for $V_{11}-\sqrt{m-\frac{p}{4}}$ and $V_{12}$ with $(m,p)=(12,5)$

Theorems & Definitions (27)

  • Definition 1.1
  • Proposition 1
  • Proposition 2
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 1
  • Definition 3.1
  • Definition 3.2
  • ...and 17 more