Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations

Tiefeng Jiang () and Danning Li ()
Additional contact information
Tiefeng Jiang: University of Minnesota
Danning Li: University of Minnesota

Journal of Theoretical Probability, 2015, vol. 28, issue 3, 804-847

Abstract: Abstract Let $$\lambda _1, \ldots , \lambda _n$$ λ 1 , … , λ n be random eigenvalues coming from the beta-Laguerre ensemble with parameter $$p$$ p , which is a generalization of the real, complex and quaternion Wishart matrices of parameter $$(n,p).$$ ( n , p ) . In the case that the sample size $$n$$ n is much smaller than the dimension of the population distribution $$p$$ p , a common situation in modern data, we approximate the beta-Laguerre ensemble by a beta-Hermite ensemble, which is a generalization of the real, complex and quaternion Wigner matrices. As corollaries, when $$n$$ n is much smaller than $$p,$$ p , we show that the largest and smallest eigenvalues of the complex Wishart matrix are asymptotically independent; we obtain the limiting distribution of the condition numbers as a sum of two i.i.d. random variables with a Tracy–Widom distribution, which is much different from the exact square case that $$n=p$$ n = p by Edelman (SIAM J Matrix Anal Appl 9:543–560, 1988); we propose a test procedure for a spherical hypothesis test. By the same approximation tool, we obtain the asymptotic distribution of the smallest eigenvalue of the beta-Laguerre ensemble. In the second part of the paper, under the assumption that $$n$$ n is much smaller than $$p$$ p in a certain scale, we prove the large deviation principles for three basic statistics: the largest eigenvalue, the smallest eigenvalue and the empirical distribution of $$\lambda _1, \ldots , \lambda _n$$ λ 1 , … , λ n , where the last large deviation is derived by using a non-standard method.

Keywords: Laguerre ensemble; Wigner ensemble; Variation norm; Large deviation; Largest eigenvalue; Smallest eigenvalue; Empirical distribution of eigenvalues; Tracy–Widom distribution; Condition number; Primary 15B52; 60B20; 60F10; Secondary 60F05; 62H10; 62H15 (search for similar items in EconPapers)
Date: 2015
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://link.springer.com/10.1007/s10959-013-0519-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:28:y:2015:i:3:d:10.1007_s10959-013-0519-7

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-013-0519-7

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().