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

Abstract

Let $X_{n}$ be $n\times N$ random complex matrices, and let $R_{n}$ and $T_{n}$ be non-random complex matrices with dimensions $n\times N$ and $n\times n$, respectively. We assume that the entries of $X_{n}$ are normalized independent random variables satisfying the Lindeberg condition, $T_{n}$ are nonnegative definite Hermitian matrices and commutative with $R_nR_n^*$, i.e., $T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n}$. The general information-plus-noise-type matrices are defined by $C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\right) \left( R_{n}+X_{n}\right) ^{*}T_{n}^{\frac{1}{2}}$. 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.

Appendix A

Lemma A.1

(Theorem A.44. of Bai and Silverstein [24]) For $n\times N$ matrices P and Q,

$$\begin{aligned} \Vert F^{PP^{*}}-F^{QQ^{*}}\Vert \le \frac{2}{n} \mathrm {rank}(P-Q). \end{aligned}$$

Lemma A.2

Bernstein inequality If $\{X_1,\dots , X_n\}$ is a sequence of independent random variables with mean zero and uniformly bounded by b, then for any $\varepsilon >0$, we have

$$\begin{aligned} \mathbf{P}(|S_n |>\varepsilon )\le 2\exp (-\frac{\varepsilon ^2}{2B_n^2 +2b\varepsilon }), \end{aligned}$$

where $S_n=\sum _{i=1}^{n} X_i$ and $B_n^2=\mathbf{E}S_n^2$.

Lemma A.3

(Lemma 2.6 of Silverstein and Bai [14]) Let $z=u+iv \in {\mathbb {C}}$ with A $n\times n$ matrix and B Hermitian matrix, and $r \in {\mathbb {C}}^{n}$. Then

$$\begin{aligned} \begin{aligned} \left|\mathrm {tr}\left( (B-zI)^{-1}-\left( B+rr^{*}-zI\right) ^{-1}\right) A \right|=\left|\frac{r^{*}(B-z I)^{-1} A(B-z I)^{-1} r}{1+r^{*}(B-z I)^{-1} r} \right|\le \frac{ \Vert A\Vert }{v}. \end{aligned} \end{aligned}$$

Lemma A.4

(Corollary A.44. of Bai and Silverstein [24]) Let A and B be two $p \times n$ matrices and the ESDs of $S = AA^*$ and ${\overline{S}} = BB^*$ be denoted by $F^S$ and $F^{{\overline{S}}}$. Then,

$$\begin{aligned} L^4(F^S, F^{{\overline{S}}}) \le \frac{2}{p^2} (\mathrm {tr}(AA^* + BB^*)) (\mathrm {tr}(A- B)(A - B)^*) . \end{aligned}$$

where L is the Levy distance between two two-dimensional distribution functions F and G defined by

$$\begin{aligned} L(F, G) = \inf \{\varepsilon : F(x-\varepsilon , y-\varepsilon )-\varepsilon \le G(x, y) \le F(x+\varepsilon , y+\varepsilon )+\varepsilon \}. \end{aligned}$$

Lemma A.5

(Lemma 2.3 of Bai and Silverstein [25]) For $X = (X_1, \dots , X_n)^T$, with $X_i$ independent, standardized, and bounded, A a $n \times n$ matrix, we have for any $p \ge 2$

$$\begin{aligned} \mathbf{E}|X^*A X - \mathrm {tr} A |^p \le \kappa _p(\mathrm {tr} AA^*)^{\frac{p}{2}}. \end{aligned}$$

where $\kappa _p$ also depends on the bound on the $X_i$.

Lemma A.6

(Sherman–Morrison formula) For $n\times n$ matrices A and $n\times 1$ vectors q and v, where A and $A+vv^{*}$ are invertible, one has

$$\begin{aligned} q^{*}(A+vv^{*})^{-1}=q^{*}A^{-1}-\dfrac{q^{*}A^{-1}v}{1+v^{*}A^{-1}v}v^{*}A^{-1}. \end{aligned}$$

When $q=v$, then

$$\begin{aligned} v^{*}(A+vv^{*})^{-1}=\dfrac{1}{1+v^{*}A^{-1}v}v^{*}A^{-1}. \end{aligned}$$

Lemma A.7

(Theorem A.5 of Bai and Silverstein [24]) If the matrix A and $A_k$, the k-th major submatrix of A of order $(n-1)$, are both nonsingular and symmetric, then

$$\begin{aligned} \mathrm {tr}(A^{-1}) - \mathrm {tr}(A_k^{-1}) = \dfrac{1+\alpha _k'A_k^{-2}\alpha _k}{a_{kk} - \alpha _k'A_k^{-1}\alpha _k}. \end{aligned}$$

where $a_kk$ is the k-th diagonal entry of A, $\alpha _k'$ is the vector obtained from the k-th row of A, by deleting the k-th entry. If A is Hermitian, then $\alpha _k'$ is replaces by $\alpha _k^*$ in equality above.

Lemma A.8

(Lemma 2.12 of [24]) Let $X_k$ be a complex martingale difference sequence with respect to the increasing $\sigma$-field $\{F_k\}$. Then, for $p > 1$,

$$\begin{aligned} \mathbf{E}|\sum X_k|^p \le \kappa _p \mathbf{E}\left( \sum |X_k|^2 \right) ^{\frac{p}{2}}. \end{aligned}$$