Random Matrix Theory for Heavy-Tailed Time Series

Heiny, Johannes

doi:10.1007/s10958-019-04191-3

Cited by 4 publications

(8 citation statements)

References 31 publications

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“…Examples admitting heavy tails being time series analysis, disordered systems, quantum field theory and more recently deep neural networks. e.g., see [4,5,6,7,8,9,10,11,12,13,14,15,16]. From a theoretical point of view the ensembles stable under matrix addition are of particular importance since they are the fixed points of their respective domains of attraction via the multivariate central limit theorem [17], see [18] for a recent work on unitarily invariant Hermitian random matrices.…”

Section: Introductionmentioning

confidence: 99%

“…There are several works [19,4,16,20,21,22,23,24,25,26,27,29,30,31,32], in physics and mathematics that have studied heavy-tailed Wigner matrices in detail (meaning matrices whose entries are independently and identically distributed along a heavy-tailed univariate probability measure). For these matrices it has been shown [20,21,24,28] that the largest eigenvalues in the heavy tail converge to Poisson statistics.…”

Section: Introductionmentioning

confidence: 99%

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Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Kieburg,

Monteleone

2021

Preprint

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We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poissonlike statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Kieburg,

Monteleone

2021

Preprint

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“…Heavy tailed random matrix ensembles naturally appear in several applications raging from statistics to physics, engineering and machine learning, e.g., see [13,38,16,11,2,9,40,30,41,39,36,37,28,45]. Therefore, it is a very intriguing question to study also random matrix ensembles that are stable with respect to matrix addition, see (1.5).…”

Section: Introductionmentioning

confidence: 99%

“…The simplest class from its construction are heavy-tailed Wigner matrices where the matrix entries are identically and independently distributed heavy tailed random variables. They were heavily studied in physics and mathematics [19,13,28,43,10,15,6,4,49,7,8,29,12,35,24]. The stable version of these matrices are then readily found by choosing ordinary stable real random variables as the real independent matrix entries.…”

Section: Introductionmentioning

confidence: 99%

A rate of convergence when generating stable invariant Hermitian random matrix ensembles

Kieburg¹,

Zhang²

2023

Preprint

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Recently, we have classified Hermitian random matrix ensembles that are invariant under the conjugate action of the unitary group and stable with respect to matrix addition. Apart from a scaling and a shift, the whole information of such an ensemble is encoded in the stability exponent determining the "heaviness" of the tail and the spectral measure that describes the anisotropy of the probability distribution. In the present work, we address the question how these ensembles can be generated by the knowledge of the latter two quantities. We consider a sum of a specific construction of identically and independently distributed random matrices that are based on Haar distributed unitary matrices and a stable random vectors. For this construction, we derive the rate of convergence in the supremums norm and show that this rate is optimal in the class of all stable invariant random matrices for a fixed stability exponent. As a consequence we also give the rate of convergence in the total variation distance.

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“…Introductory Remarks. Heavy-tailed random matrices have been of interest, lately, since they have applications in machine learning [61,62], disordered systems [18,12], quantum field theory [46], finance [23,24,63,3,67] and statistics in general [20,42,40,41,68,66]. Especially, Wigner matrices with heavy-tailed and independently distributed matrix entries have been studied in detail [30,76,10,22,14,25,13,7,80,15,78,6,19,59,69].…”

mentioning

confidence: 99%

Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

Kieburg¹,

Zhang²

2021

Preprint

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We consider random matrix ensembles on the Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the eigenvectors are Haar distributed and, therefore, factorise from the eigenvalue statistics. We prove a classification for stable matrix ensembles of this kind of matrices represented in terms of matrices, their eigenvalues and their diagonal entries with the help of the classification of the multivariate stable distributions and the harmonic analysis on symmetric matrix spaces. Moreover, we identify sufficient and necessary conditions for their domains of attraction. To illustrate our findings we discuss for instance elliptical invariant random matrix ensembles and Pólya ensembles. As a byproduct we generalise the derivative principle on the Hermitian matrices to general tempered distributions.

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Random Matrix Theory for Heavy-Tailed Time Series

Cited by 4 publications

References 31 publications

Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

A rate of convergence when generating stable invariant Hermitian random matrix ensembles

Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles

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