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

• Published in: arXiv.org
• Main Authors: Chen, B B, Pan, G M
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.11.2012
• Subjects: Convergence; Covariance matrix; Eigenvalues; Mathematics - Statistics Theory; Random variables; Statistics - Theory
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1211.5479

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 \(EX_{11}=0,EX_{11}^2=1\) and \(EX_{11}^4<\infty\). 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 \(p\rightarrow\infty,n\rightarrow\infty\) with \(p/n\rightarrow0\).