Operator-valued matrices with free or exchangeable entries

Marwa Banna; Guillaume Cébron

doi:10.1214/22-AIHP1255

Operator-valued matrices with free or exchangeable entries

Marwa Banna, Guillaume Cébron

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study matrices whose entries are free or exchangeable noncommutative elements in some tracial W^∗-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and prove quantitative convergence to operator-valued semicircular elements over some subalgebra in terms of Cauchy transforms and the Kolmogorov distance. As direct applications, we obtain explicit rates of convergence for a large class of random block matrices with independent or correlated blocks. Our approach relies on a noncommutative extension of the Lindeberg method and operator-valued Gaussian interpolation techniques.

Original language	English (US)
Pages (from-to)	503-537
Number of pages	35
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	59
Issue number	1
DOIs	https://doi.org/10.1214/22-AIHP1255
State	Published - Feb 2023

Keywords

Matrices with free entries
Matrices with noncommutative exchangeable entries
Noncommutative Lindeberg method
Operator-valued free probability
Random block matrices
Random operators

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1214/22-AIHP1255

Cite this

@article{a85c6424d9e946f289e74f3a4c54e8b5,

title = "Operator-valued matrices with free or exchangeable entries",

abstract = "We study matrices whose entries are free or exchangeable noncommutative elements in some tracial W∗-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and prove quantitative convergence to operator-valued semicircular elements over some subalgebra in terms of Cauchy transforms and the Kolmogorov distance. As direct applications, we obtain explicit rates of convergence for a large class of random block matrices with independent or correlated blocks. Our approach relies on a noncommutative extension of the Lindeberg method and operator-valued Gaussian interpolation techniques.",

keywords = "Matrices with free entries, Matrices with noncommutative exchangeable entries, Noncommutative Lindeberg method, Operator-valued free probability, Random block matrices, Random operators",

author = "Marwa Banna and Guillaume C{\'e}bron",

note = "Publisher Copyright: {\textcopyright} Association des Publications de l{\textquoteright}Institut Henri Poincar{\'e}, 2023.",

year = "2023",

month = feb,

doi = "10.1214/22-AIHP1255",

language = "English (US)",

volume = "59",

pages = "503--537",

journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "1",

}

TY - JOUR

T1 - Operator-valued matrices with free or exchangeable entries

AU - Banna, Marwa

AU - Cébron, Guillaume

N1 - Publisher Copyright: © Association des Publications de l’Institut Henri Poincaré, 2023.

PY - 2023/2

Y1 - 2023/2

N2 - We study matrices whose entries are free or exchangeable noncommutative elements in some tracial W∗-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and prove quantitative convergence to operator-valued semicircular elements over some subalgebra in terms of Cauchy transforms and the Kolmogorov distance. As direct applications, we obtain explicit rates of convergence for a large class of random block matrices with independent or correlated blocks. Our approach relies on a noncommutative extension of the Lindeberg method and operator-valued Gaussian interpolation techniques.

AB - We study matrices whose entries are free or exchangeable noncommutative elements in some tracial W∗-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and prove quantitative convergence to operator-valued semicircular elements over some subalgebra in terms of Cauchy transforms and the Kolmogorov distance. As direct applications, we obtain explicit rates of convergence for a large class of random block matrices with independent or correlated blocks. Our approach relies on a noncommutative extension of the Lindeberg method and operator-valued Gaussian interpolation techniques.

KW - Matrices with free entries

KW - Matrices with noncommutative exchangeable entries

KW - Noncommutative Lindeberg method

KW - Operator-valued free probability

KW - Random block matrices

KW - Random operators

UR - http://www.scopus.com/inward/record.url?scp=85152145408&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85152145408&partnerID=8YFLogxK

U2 - 10.1214/22-AIHP1255

DO - 10.1214/22-AIHP1255

M3 - Article

AN - SCOPUS:85152145408

SN - 0246-0203

VL - 59

SP - 503

EP - 537

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 1

ER -

Operator-valued matrices with free or exchangeable entries

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this