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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.

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Acknowledgements

Bai’s research was supported by NSFC No. 12171198 and STDFJ No. 20210101147JC. Hu’s research was supported by NSFC No. 12171078, 11971097 and National Key R &D Program of China No. 2020YFA0714102.

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Appendix A

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 \( p > 1 \),

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

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Zhou, H., Bai, Z. & Hu, J. The Limiting Spectral Distribution of Large-Dimensional General Information-Plus-Noise-Type Matrices. J Theor Probab (2022). https://doi.org/10.1007/s10959-022-01193-x

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  • DOI: https://doi.org/10.1007/s10959-022-01193-x

Keywords

  • Limiting spectral distribution
  • Random matrix theory
  • Stieltjes transform

Mathematics Subject Classification

  • 60B20
  • 62H10