Products of Independent Elliptic Random Matrices

Consider the product X = X 1 · · · Xm of m independent n × n iid random matrices. When m is fixed and the dimension n tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of X for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on m (the number of factor matrices) or on the distribution of the entries of the matrices. The main result generalizes and improves upon previous limit statements for the linear spectral statistics of a single iid matrix by Rider and Silverstein as well as Renfrew and the second author.

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“…Similar results are stated in [33]. Theorem 1.3 (Theorem 2.4 from [53]). Let m ≥ 1 be an integer and τ > 0.…”

Section: Introduction and Background Materialssupporting

confidence: 86%

Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices

Coston

O’Rourke

2019

J Theor Probab

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“…Since the paper of [1] it was used many times (see, e.g., [7], [15] and [13]), but the method of the proof of CLT used in the present paper allows to prove CLT by the same way for all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse and diluted random matrices etc. It becomes even simpler than that for band matrices, since the proof of condition (2) becomes simpler.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Fluctuations of Eigenvalues of Random Band Matrices

Shcherbina

2015

J Stat Phys

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We consider the fluctuation of linear eigenvalue statistics of random band n×n matrices whose entries have the form Mij = b −1/2 u 1/2 (|i − j|)wij with i.i.d. wij possessing the (4 + ε)th moment, where the function u has a finite support [−C * , C * ], so that M has only 2C * b+1 nonzero diagonals. The parameter b (called the bandwidth) is assumed to grow with n in a way that b/n → 0. Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we improve and generalize the results of the previous papers [8] and [11], where CLT was proven under the assumption n >> b >> n 1/2 . Moreover, we develop a method which allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc.

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“…The following proposition relates the eigenvalues of M to the eigenvalues of the product M 1 · · · M m . We note that similar linearization tricks have been used previously; see, for example, [9,30,41,50,51] and references therein. Proof.…”

Section: 2mentioning

confidence: 99%

“…In this paper, we focus on the product of several independent iid matrices. In this case, the analogue of the circular law (Theorem 1.2) is the following result from [50], due to Renfrew, Soshnikov, Vu, and the second author of the current paper. Theorem 1.6 (Theorem 2.4 from [50]).…”

Section: Introductionmentioning

confidence: 96%

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Outliers in the spectrum for products of independent random matrices

Coston¹,

O’Rourke²,

Wood³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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For fixed m ≥ 1, we consider the product of m independent n × n random matrices with iid entries as n → ∞. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Tao [60] for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. 42 12. Proofs of results from Section 3 54 Appendix

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Products of Independent Elliptic Random Matrices

Cited by 36 publications

References 58 publications

Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices

Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices

On Fluctuations of Eigenvalues of Random Band Matrices

Outliers in the spectrum for products of independent random matrices

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