Central limit theorem for products of random matrices

Berger, Marc A.

doi:10.1090/s0002-9947-1984-0752503-3

Cited by 28 publications

(3 citation statements)

References 18 publications

(10 reference statements)

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“…When applied to the overall product matrix M (t) , if all gain vectors g (k) are small, the matrix M (t) is the exponential of the Gaussian ensemble. The approximation used here is similar to the results of Berger [123]. This small-gain approximation yields Proposition I, but not Proposition II.…”

Section: Properties Of the Product Of Random Matricessupporting

confidence: 62%

Optical Fiber Telecommunications VIB

2013

Optical Fiber Telecommunications

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Section: Properties Of the Product Of Random Matricessupporting

confidence: 62%

Optical Fiber Telecommunications VIB

2013

Optical Fiber Telecommunications

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“…For instance, sums and products of random matrices have been used in the context of multiple channel communication [5,6,[27][28][29][30][31], quantum entanglement problem [47], neutral network analyses [49,50], and random supergravity theories [51][52][53]. Product of random matrices have also found applications in problems related to stochastic differential equations and Lyapunov exponents [48,[54][55][56][57][58], fixed point analysis for multi-layered complex systems [38], and Markov chains with random transition probabilities [39].…”

Section: Introductionmentioning

confidence: 99%

Spectral statistics for the difference of two Wishart matrices

Kumar

Charan

2020

J. Phys. A: Math. Theor.

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In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean.

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“…Product of random matrices and the associated eigenvalue spectra exhibit a number of interesting properties and find concrete applications in varied fields of knowledge. Their study goes back to as early as the 1950s where the focus was on exploring the behavior of dynamical systems, and the accompanying questions related to stochastic differential equations and Lyapunov exponents [1][2][3][4][5]. In the last few years there has been a revival in interest in their investigation because of their fascinating integrability properties [6][7][8][9][10][11][12][13][14][15][16][17][18][19] and identification of new problems where they can be applied, such as random graph states [20], combinatorics [21], quantum entanglement [14] and multilayered multiple channel telecommunication [9,22].…”

Section: Introductionmentioning

confidence: 99%