No eigenvalues outside the support of the limiting spectral distribution of quaternion sample covariance matrices

In this paper, we consider the spectral properties of quaternion sample covariance matrices. Let [Formula: see text], where [Formula: see text] is the square root of a [Formula: see text] quaternion Hermitian non-negative definite matrix [Formula: see text] and [Formula: see text] is a [Formula: see text] matrix consisting of i.i.d. standard quaternion entries. Under the framework of random matrix theory, i.e. [Formula: see text] as [Formula: see text], we prove that if the fourth moment of the entries is finite, then there will almost surely be no eigenvalues that appear in any closed interval outside the support of the limiting distribution as [Formula: see text].