Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero

Abstract

Let ${\mathbf{X}}_p=({\mathbf{s}}_1,...,{\mathbf{s}}_n)=(X_{ij}{)}_{p\times n}$ where $X_{ij}$'s are independent and identically distributed (i.i.d.) random variables with $E{X}_{11}=0,E{X}_{11}^2=1$ and $E{X}_{11}^4<\infty$. It is showed that the largest eigenvalue of the random matrix ${\mathbf{A}}_p=\frac{1}{2\sqrt{np}}({\mathbf{X}}_p{\mathbf{X}}_p^{\prime }-n{\mathbf{I}}_p)$ tends to 1 almost surely as $p\to \infty ,n\to \infty$ with $p/n\to 0$.

Published in at http://dx.doi.org/10.3150/11-BEJ381 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles
• On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity

Cited by in corpus (4)

• CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size
• On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime
• Asymptotic normality for eigenvalue statistics of a general sample covariance matrix when $p/n\to \infty$ and applications
• Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks