Convergence of empirical spectral distributions of large dimensional quaternion sample covariance matrices

In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices. Motivated by Bai and Silverstein (Spectral analysis of large dimensional random matrices, Springer, New York, 2010 ) and Marčenko and Pastur (Matematicheskii Sb, 114:507–536, 1967 ), we can extend the results of the real or complex sample covariance matrix to the quaternion case. Suppose X n = ( x j k ( n ) ) p × n is a quaternion random matrix. For each n , the entries { x i j ( n ) } are independent random quaternion variables with a common mean μ and variance σ 2 > 0 . It is shown that the empirical spectral distribution of the quaternion sample covariance matrix S n = n - 1 X n X n ∗ converges to the Marčenko–Pastur law as p → ∞ , n → ∞ and p / n → y ∈ ( 0 , + ∞ ) .