The limiting spectral distribution of large dimensional general information-plus-noise type matrices

01/28/2022

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by   Huanchao Zhou, et al.

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Let X_n be n× N random complex matrices, R_n and T_n be non-random complex matrices with dimensions n× N and n× n, respectively. We assume that the entries of X_n are independent and identically distributed, T_n are nonnegative definite Hermitian matrices and T_nR_nR_n^*= R_nR_n^*T_n. The general information-plus-noise type matrices are defined by C_n=1/NT_n^1/2( R_n +X_n) (R_n+X_n)^*T_n^1/2. In this paper, we establish the limiting spectral distribution of the large dimensional general information-plus-noise type matrices C_n. Specifically, we show that as n and N tend to infinity proportionally, the empirical distribution of the eigenvalues of C_n converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform.