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Large deviations of extremal eigenvalues of sample covariance matrices
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2023 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 60, no 4, p. 1275-1280Article in journal (Refereed) Published
Abstract [en]

Large deviations of the largest and smallest eigenvalues of XX/n are studied in thisnote, where Xp×n is a p × n random matrix with independent and identically distributed(i.i.d.) sub-Gaussian entries. The assumption imposed on the dimension size p andthe sample size n is p = p(n) → ∞ with p(n) = o(n). This study generalizes one resultobtained in [3].

Place, publisher, year, edition, pages
Cambridge University Press, 2023. Vol. 60, no 4, p. 1275-1280
Keywords [en]
Large deviations; sample covariance matrices; extremal eigenvalues
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-193671DOI: 10.1017/jpr.2022.130ISI: 000977978400001OAI: oai:DiVA.org:liu-193671DiVA, id: diva2:1756503
Available from: 2023-05-12 Created: 2023-05-12 Last updated: 2024-10-01Bibliographically approved
In thesis
1. Large deviations of condition numbers and extremal eigenvalues of random matrices
Open this publication in new window or tab >>Large deviations of condition numbers and extremal eigenvalues of random matrices
2023 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Random matrix theory found applications in many areas, for instance in statistics random matrices are used to analyse multivariate data and their eigenvalues are used in hypothesis testing. Spectral properties of random matrices have been studied extensively in the literature dealing with both the bulk case (involving all the eigenvalues) and the extremal case (addressing the maximal and minimal eigenvalues). In this thesis two types of sequences of random matrices are considered: the first type is the sequence of sample covariance matrices, and the second type is the sequence of β-Laguerre (or Wishart) ensembles, for which large deviations of their extremal cases are studied. These two types of sequences of random matrices contain the classical Wishart matrices.

The thesis can be divided into two parts. The first part is on the study of large deviations of condition numbers defined as ratios of maximal and the minimal eigenvalues. This is done based on suitable analysis and estimates of the joint density function of all eigenvalues. The second part deals with large deviations of individual maximal and minimal eigenvalue, and the approach consists of suitable eigenvalue concentration inequalities and Laplace’s method.

It is remarked that for those two types of sequences of random matrices considered in this thesis, two scenarios are investigated: either one of the dimension size and the sample size is much larger than the other one, or the two sizes are comparable.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2023. p. 27
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2313
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-193345 (URN)10.3384/9789180751575 (DOI)9789180751568 (ISBN)9789180751575 (ISBN)
Public defence
2023-06-01, C2, C Building, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Note

Funding agency: Swedish International Development Cooperation Agency (Sida) through Rwanda-Sweden bilateral programme

Available from: 2023-05-02 Created: 2023-05-02 Last updated: 2023-05-12Bibliographically approved

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Uwamariya, DeniseYang, Xiangfeng

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