Spectral Analysis of Large Dimensional Random Matrices

Bai, Zhidong; Silverstein, Jack W.

doi:10.1007/978-1-4419-0661-8

Cited by 1,322 publications

(2,139 citation statements)

References 137 publications

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“…This is a direct consequence of a recent result by Jiang and Li [21]. The situation for q/p → c > 0 is well known; see, for example, [3]. (5) Connections in (1.2) and (1.3) among distances provide an efficient way to use known properties of Wishart matrices and Haar-invariant orthogonal matrices.…”

Section: Remarks and Future Questionsmentioning

confidence: 85%

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Jiang

2019

Trans. Amer. Math. Soc.

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Abstract. Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √ nZn and Gn in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq 2 /n goes to zero, and not so if (p, q) sits on the curve pq 2 = σn. A previous work by Jiang [17] shows that the total variation distance goes to zero if both p/ √ n and q/ √ n go to zero, and it is not true provided p = c √ n and q = d √ n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ.

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Section: Remarks and Future Questionsmentioning

confidence: 85%

“…The CLT for the case q/p → c > 0 is well known; see, for example, [3] or [22]. (4) Some properties of the largest and the smallest eigenvalues of G ′ n G n for the case q/p → 0 is proved in Lemma 2.9.…”

Section: Remarks and Future Questionsmentioning

confidence: 99%

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Jiang

2019

Trans. Amer. Math. Soc.

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“…where √ z 2 − 4 is the branch of square root with a branch cut in [−2, 2] and asymptotically equals z at infinity [4].…”

Section: Random Matrices and The Stieltjes Transformmentioning

confidence: 99%

“…If the Stieltjes transforms of two spectral measures are close to each other (for all z), then the two measures are more or less the same. In particular, if s n (z) is close to s sc (z), then the spectral distribution of W n is close to the semi-circle distribution (see for instance [4,Chapter 11], [10]). We are going to use the following lemma.…”

Section: Random Matrices and The Stieltjes Transformmentioning

confidence: 99%

Random weighted projections, random quadratic forms and random eigenvectors

Wang

2014

Random Struct. Alg.

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“…In order to establish thatμ has a deterministic behaviour when t → +∞, it is well known that it is sufficient to establish that (3) holds for φ(λ) = 1 λ+σ 2 for each σ 2 > 0 (see e.g., [19,20]). In order to reformulate (3) for φ(λ) = 1 λ+σ 2 , we introduce the Stieltjes transform s ν of a positive measure ν carried by R + , which is the function defined for each z ∈ C − R − as:…”

Section: The Case Q = Imentioning

confidence: 99%

Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

Dupuy

Loubaton²

2012

Entropy

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This paper provides a comprehensive introduction of large random matrix methods for input covariance matrix optimization of mutual information of MIMO systems. It is first recalled informally how large system approximations of mutual information can be derived. Then, the optimization of the approximations is discussed, and important methodological points that are not necessarily covered by the existing literature are addressed, including the strict concavity of the approximation, the structure of the argument of its maximum, the accuracy of the large system approach with regard to the number of antennas, or the justification of iterative water-filling optimization algorithms. While the existing papers have developed methods adapted to a specific model, this contribution tries to provide a unified view of the large system approximation approach.

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Spectral Analysis of Large Dimensional Random Matrices

Cited by 1,322 publications

References 137 publications

Plot of CLT

Plot of CLT

Random weighted projections, random quadratic forms and random eigenvectors

Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

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