Joint CLT for several random sesquilinear forms with applications to large-dimensional spiked population models. (English) Zbl 1307.60017

Wang, Qinwen; Su, Zhonggen; Yao, Jianfeng

Electron. J. Probab. 19, Paper No. 103, 28 p. (2014).

Summary: In this paper, we derive a joint central limit theorem for random vectors whose components are functions of random sesquilinear forms. This result is a natural extension of the existing central limit theory on random quadratic forms. We also provide applications in random matrix theory related to large-dimensional spiked population models. For the first application, we find the joint distribution of grouped extreme sample eigenvalues corresponding to the spikes. For the second application, under the assumption that the population covariance matrix is diagonal with \(k\) (fixed) simple spikes, we derive the asymptotic joint distribution of the extreme sample eigenvalue and its corresponding sample eigenvector projection.

MSC:

60F05	Central limit and other weak theorems
60B20	Random matrices (probabilistic aspects)

Keywords:

central limit theorem; random vectors; random sesquilinear forms; random matrices; extreme eigenvalues; extreme eigenvectors; joint distribution; spiked population model