Asymptotic Normality of the Largest Eigenvalue for Noncentral Sample Covariance Matrices
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913572501585920 |
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| author | Cheng, Huihui Song, Minjie |
| author_facet | Cheng, Huihui Song, Minjie |
| contents | Let $X$ be a $p\times n$ independent identically distributed real Gaussian matrix with positive mean $μ$ and variance $σ^2$ entries. The goal of this paper is to investigate the largest eigenvalue of the noncentral sample covariance matrix $W=XX^{T}/n$, when the dimension $p$ and the sample size $n$ both grow to infinity with the limit $p/n=c\,(0<c<\infty)$. Utilizing the von Mises iteration method, we derive an approximation of the largest eigenvalue $λ_{1}(W)$ and show that $λ_{1}(W)$ asymptotically has a normal distribution with expectation $pμ^2+(1+c)σ^2$ and variance $4cμ^2σ^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Normality of the Largest Eigenvalue for Noncentral Sample Covariance Matrices Cheng, Huihui Song, Minjie Probability Let $X$ be a $p\times n$ independent identically distributed real Gaussian matrix with positive mean $μ$ and variance $σ^2$ entries. The goal of this paper is to investigate the largest eigenvalue of the noncentral sample covariance matrix $W=XX^{T}/n$, when the dimension $p$ and the sample size $n$ both grow to infinity with the limit $p/n=c\,(0<c<\infty)$. Utilizing the von Mises iteration method, we derive an approximation of the largest eigenvalue $λ_{1}(W)$ and show that $λ_{1}(W)$ asymptotically has a normal distribution with expectation $pμ^2+(1+c)σ^2$ and variance $4cμ^2σ^2$. |
| title | Asymptotic Normality of the Largest Eigenvalue for Noncentral Sample Covariance Matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2410.17085 |