The Limiting Spectral Distribution of Large-Dimensional General Information-Plus-Noise-Type Matrices

Let $$ X_{n} $$ X n be $$ n\times N $$ n × N random complex matrices, and let $$R_{n}$$ R n and $$T_{n}$$ T n be non-random complex matrices with dimensions $$n\times N$$ n × N and $$n\times n$$ n × n , respectively. We assume that the entries of $$ X_{n} $$ X n are normalized independent random variables satisfying the Lindeberg condition, $$ T_{n} $$ T n are nonnegative definite Hermitian matrices and commutative with $$R_nR_n^*$$ R n R n ∗ , i.e., $$T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n} $$ T n R n R n ∗ = R n R n ∗ T n . The general information-plus-noise-type matrices are defined by $$C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\right) \left( R_{n}+X_{n}\right) ^{*}T_{n}^{\frac{1}{2}} $$ C n = 1 N T 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 $$C_{n}$$ C n . Specifically, we show that as n and N tend to infinity proportionally, the empirical distribution of the eigenvalues of $$C_{n}$$ C n converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform.