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<p>A random matrix is called a Gaussian matrix if its vectorization follows a Gaussian distribution. This paper investigates the covariances of random matrices and symmetric random matrices. We introduce the concepts of the Gaussian matrix, line normal matrix, and covariance tensors, and study the positive definiteness of the 4-order tensors. We show that the covariance tensors are positive definite, and characterize the covariance tensors of the standard normal distributed random matrices. We also use tensors to simplify the expression of the <i>k</i>-moments of random vectors and random matrices, and shows that all the <i>k</i>-moments of a centralized random vector with Gaussian distribution is zero if <i>k</i> is an odd number.</p>

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Fourth Order Random Tensors and Their Applications in Statistics

  • 期刊论文

  • 发表日期：

    2025年5月12日

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  • Ziqi Zhai,
  • Li Wang,
  • Changqing Xu

摘要

A random matrix is called a Gaussian matrix if its vectorization follows a Gaussian distribution. This paper investigates the covariances of random matrices and symmetric random matrices. We introduce the concepts of the Gaussian matrix, line normal matrix, and covariance tensors, and study the positive definiteness of the 4-order tensors. We show that the covariance tensors are positive definite, and characterize the covariance tensors of the standard normal distributed random matrices. We also use tensors to simplify the expression of the k-moments of random vectors and random matrices, and shows that all the k-moments of a centralized random vector with Gaussian distribution is zero if k is an odd number.