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

Huanchao Zhou, Zhidong Bai and Jiang Hu ()
Additional contact information
Huanchao Zhou: Northeast Normal University
Zhidong Bai: Northeast Normal University
Jiang Hu: Northeast Normal University

Journal of Theoretical Probability, 2023, vol. 36, issue 2, 1203-1226

Abstract: Abstract 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.

Keywords: Limiting spectral distribution; Random matrix theory; Stieltjes transform; 60B20; 62H10 (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-022-01193-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:36:y:2023:i:2:d:10.1007_s10959-022-01193-x

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-022-01193-x

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().