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

Kieburg, Mario; Zhang, Jiyuan

doi:10.48550/arxiv.2110.14877

Cited by 2 publications

(3 citation statements)

References 81 publications

(128 reference statements)

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“…To understand this and other peculiarities for invariant stable random matrices, we have started an approach from finite matrix dimension. In a recent work [34], we have classified all unitarily invariant stable random matrix ensembles on the Hermitian matrices Herm(N ) and we have identified their domains of attraction meaning we have proven central limit theorems. In the next step, which is the present work about, we would like to show how those invariant stable random matrix ensembles can be created via an approximation and how fast the convergence of this approximation is.…”

Section: Introductionmentioning

confidence: 99%

“…In a previous work of the authors [34], the stable distributions for unitarily invariant Hermitian random matrices are introduced. They are special cases of multivariate stable distributions, with the extra invariance condition that when applying a unitary similarity transformation the distribution is unchanged.…”

Section: Introductionmentioning

confidence: 99%

“…In Sec. 1.2 we will firstly re-state the classification result of stable invariant ensembles introduced in [34]. Then we present three main results: Theorem 1 states the regularity of stable invariant ensembles; Theorem 3 suggests an approximation and states its rate of convergence in terms of the supremum norm; and Corollary 18 states the rate of convergence of the total variation distance.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kieburg¹,

Zhang²

2023

Preprint

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Derivative principles for invariant ensembles

Kieburg¹,

Zhang²

2020

Preprint

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A big class of ensembles of random matrices are those which are invariant under group actions so that the eigenvectors become statistically independent of the eigenvalues. In the present work we show that this is not the sole consequence but that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities which are sometimes easier accessible. Those quantities might be the diagonal matrix entries as it is the case for Hermitian matrices or some pseudo-diagonal entries for other matrix spaces. These representations are called derivative principles. We show them for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. In all six cases we prove a uniqueness of the derivative principle as it is not trivial to see that the derivative operator has a trivial kernel on the selected spaces of the identified matrix quantities.

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

Cited by 2 publications

References 81 publications

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

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

Derivative principles for invariant ensembles

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