Another look at the moment method for large dimensional random matrices

Bose, Arup; Sen, Arnab

doi:10.1214/ejp.v13-501

Cited by 46 publications

(114 citation statements)

References 14 publications

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“…Bryc et al (2006) used combinatorial methods to prove the existence of limiting ESDs for random matrices of Hankel, Toeplitz and Markov types. Further theoretical developments for such matrices were made by Bose and Sen (2008). Recently, El Karoui (2009a), Zhang (2006), under slightly different assumptions, have proved the existence of the limiting ESD for matrices of the form…”

Section: Extension To Non-iid Settingsmentioning

confidence: 99%

Random matrix theory in statistics: A review

Paul

Aue

2014

Journal of Statistical Planning and Inference

163

115

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b s t r a c tWe give an overview of random matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the development of the theory and practice in high-dimensional statistical inference has been influenced by the corresponding developments in the field of RMT.

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Section: Extension To Non-iid Settingsmentioning

confidence: 99%

Random matrix theory in statistics: A review

Paul

Aue

2014

Journal of Statistical Planning and Inference

163

115

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“…Let us explain briefly what they mean by ''random Markov matrices''. They proved the following theorem (see [19, and also [20]) : let (X i,j ) 1<i<j<∞ be an infinite triangular array of i.i.d. real random variables of mean 0 and variance 1.…”

Section: Random Q -Matricesmentioning

confidence: 99%

The Dirichlet Markov Ensemble

Chafäı

2010

Journal of Multivariate Analysis

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We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if $\bM$ is such a random matrix, then the empirical distribution built from the singular values of$\sqrt{n} \bM$ tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of $\sqrt{n} \bM$ tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of $\bM$ is of order $1-1/\sqrt{n}$ when $n$ is large.Comment: Improved version. Accepted for publication in JMV

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“…Denote by C L h,3+ the set of all L-matched circuits of length h with at least one edge of order ≥ 3. If L satisfies Property B, then Lemma 1(a) of [7] ensures that…”

Section: Bulk Behavior Of Schur-hadamard Productsmentioning

confidence: 99%

“…the case where X n is Toeplitz and Y n is Hankel (see the second row of Table 2). In order to proceed further we need the concept of Catalan words from [7].…”

Section: Toeplitz and Hankelmentioning

confidence: 99%

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Bulk behavior of Schur–Hadamard products of symmetric random matrices

Bose

Mukherjee

2014

Random Matrices: Theory Appl.

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We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur-Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSD of Schur-Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur-Hadamard product of independent Toeplitz and Hankel matrices has the semicircular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.

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Another look at the moment method for large dimensional random matrices

Cited by 46 publications

References 14 publications

Random matrix theory in statistics: A review

Random matrix theory in statistics: A review

The Dirichlet Markov Ensemble

Bulk behavior of Schur–Hadamard products of symmetric random matrices

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