English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let \(\lambda _1, \ldots , \lambda _n\) be random eigenvalues coming from the beta-Laguerre ensemble with parameter \(p\), which is a generalization of the real, complex and quaternion Wishart matrices of parameter \((n,p).\) In the case that the sample size \(n\) is much smaller than the dimension of the population distribution \(p\), 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 \(n\) is much smaller than \(p,\) 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 \(n=p\) 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 \(n\) is much smaller than \(p\) 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 \(\lambda _1, \ldots , \lambda _n\), where the last large deviation is derived by using a non-standard method.

Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations