Recent advances on eigenvalues of matrix-valued stochastic processes

Brownian sheets; Dyson Brownian motion; Eigenvalue distribution; Fractional Brownian motion; Matrix-valued process; Squared Bessel particle system; Wishart process; Statistics and Probability; Numerical Analysis; Statistics, Probability and Uncertainty

Abstract :

[en] Since the introduction of Dyson's Brownian motion in early 1960s, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great detail. In particular, the limiting behaviours of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most recent variations of such processes, such as matrix-valued processes driven by fractional Brownian motion or Brownian sheet, the eigenvalues of them are also discussed in this survey. In the end, some open problems in the area are also proposed.

Disciplines :

Mathematics

Author, co-author :

Song, Jian ; Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, China

Yao, Jianfeng ; Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong

YUAN, Wangjun ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; Department of Mathematics and Statistics, University of Ottawa, Canada

External co-authors :

yes

Language :

English

Title :

Recent advances on eigenvalues of matrix-valued stochastic processes

Publication date :

March 2022

Journal title :

Journal of Multivariate Analysis

ISSN :

0047-259X

eISSN :

1095-7243

Publisher :

Academic Press Inc.

Volume :

188

Pages :

104847

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0047259X21001251?httpAccept=text/xml

Funding text :

The authors would like to thank the Editor, Prof. Dietrich von Rosen, for his invitation to contribute to this special jubilee issue. Comments from Prof. Victor Pérez-Abreu on this review are also acknowledged. Jian Song’s research is partially supported by Shandon University (Grant No. 11140089963041 ) and the National Natural Science Foundation of China (Grant No. 12071256 ). Jianfeng Yao’s research is partially supported by the Research Grant Council of Hong Kong SAR (GRF Grant 17307319 ).

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