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G. M. Pan

Publications and source records attributed to G. M. Pan.

8 recordsLinked to original sources
arXiv2018

Estimating a change point in a sequence of very high-dimensional covariance matrices

This paper considers the problem of estimating a change point in the covariance matrix in a sequence of high-dimensional vectors, where the dimension is substantially larger than the sample size. A two-stage approach is proposed to efficiently estimate the location of the change point. The first step consists of a reduction of the dimension to identify elements of the covariance matrices corresponding to significant changes. In a second step we use the components after dimension reduction to determine the position of the change point. Theoretical properties are developed for both steps and numerical studies are conducted to support the new methodology.

stat.ME↗
arXiv2018

Successive X-class flares and coronal mass ejections driven by shearing motion and sunspot rotation in active region NOAA 12673

We present a clear case study on the occurrence of two successive X-class flares including a decade-class flare (X9.3) and two coronal mass ejections (CMEs) triggered by shearing motion and sunspot rotation in active region NOAA 12673 on 2017 September 6. A shearing motion between the main sunspots with opposite polarities started on September 5 and even lasted after the second X-class flare on September 6. Moreover, the main sunspot with negative polarity rotated around its umbral center and another main sunspot with positive polarity also exhibited a slow rotation. The sunspot with negative polarity at the northwest of active region also began to rotate counter-clockwise before the onset of the first X-class flare. The successive formation and eruption of two S-shaped structures were closely related to the counter-clockwise rotation of three sunspots. It is also found that the rotation of sunspots is faster during four hours prior to the onset of the flares than the period before. The existence of a flux rope is found prior to the onset of two flares by using non-linear force free field extrapolation based on the vector magnetograms observed by SDO/HMI. These results suggest that shearing motion and sunspot rotation play an important role in the buildup of the free energy and the formation of flux ropes in the corona which produces solar flares and CMEs.

astro-ph.SR↗
arXiv2015

The Tracy-Widom law for the Largest Eigenvalue of F Type Matrix

Let $\mathbb{A}_p=\frac{\mathbb{Y}\mathbb{Y}^*}{m}$ and $\mathbb{B}_p=\frac{\mathbb{X}\mathbb{X}^*}{n}$ be two independent random matrices where $\mathbb{X}=(X_{ij})_{p \times n}$ and $\mathbb{Y}=(Y_{ij})_{p \times m}$ respectively consist of real (or complex) independent random variables with $\mathbb{E}X_{ij}=\mathbb{E}Y_{ij}=0$, $\mathbb{E}|X_{ij}|^2=\mathbb{E}|Y_{ij}|^2=1$. Denote by $λ_{1}$ the largest root of the determinantal equation $\det(λ\mathbb{A}_p-\mathbb{B}_p)=0$. We establish the Tracy-Widom type universality for $λ_{1}$ under some moment conditions on $X_{ij}$ and $Y_{ij}$ when $p/m$ and $p/n$ approach positive constants as $p\rightarrow\infty$.

math.ST↗
arXiv2012

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

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 $p/n\rightarrow0$.

math.ST↗
arXiv2012

Independence Test for High Dimensional Random Vectors

This paper proposes a new mutual independence test for a large number of high dimensional random vectors. The test statistic is based on the characteristic function of the empirical spectral distribution of the sample covariance matrix. The asymptotic distributions of the test statistic under the null and local alternative hypotheses are established as dimensionality and the sample size of the data are comparable. We apply this test to examine multiple MA(1) and AR(1) models, panel data models with some spatial cross-sectional structures. In addition, in a flexible applied fashion, the proposed test can capture some dependent but uncorrelated structures, for example, nonlinear MA(1) models, multiple ARCH(1) models and vandermonde matrices. Simulation results are provided for detecting these dependent structures. An empirical study of dependence between closed stock prices of several companies from New York Stock Exchange (NYSE) demonstrates that the feature of cross--sectional dependence is popular in stock markets.

math.ST↗
arXiv2008

Central limit theorem for signal-to-interference ratio of reduced rank linear receiver

Let $\mathbf{s}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ with $\{v_{ik},i,k=1,...\}$ independent and identically distributed complex random variables. Write $\mathbf{S}_k=(\mathbf{s}_1,...,\mathbf {s}_{k-1},\mathbf{s}_{k+1},... ,\mathbf{s}_K),$ $\mathbf{P}_k=\operatorname {diag}(p_1,...,p_{k-1},p_{k+1},...,p_K)$, $\mathbf{R}_k=(\mathbf{S}_k\mathbf{P}_k\mathbf{S}_k^*+σ^2\mathbf{I})$ and $\mathbf{A}_{km}=[\mathbf{s}_k,\mathbf{R}_k\mathbf{s}_k,... ,\mathbf{R}_k^{m-1}\mathbf{s}_k]$. Define $β_{km}=p_k\mathbf{s}_k^*\mathbf{A}_{km}(\mathbf {A}_{km}^*\times\ mathbf{R}_k\mathbf{A}_{km})^{-1}\mathbf{A}_{km}^*\mathbf{s}_k$, referred to as the signal-to-interference ratio (SIR) of user $k$ under the multistage Wiener (MSW) receiver in a wireless communication system. It is proved that the output SIR under the MSW and the mutual information statistic under the matched filter (MF) are both asymptotic Gaussian when $N/K\to c>0$. Moreover, we provide a central limit theorem for linear spectral statistics of eigenvalues and eigenvectors of sample covariance matrices, which is a supplement of Theorem 2 in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532--1572]. And we also improve Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553--605].

math.PR↗
arXiv2007

On asymptotics of eigenvectors of large sample covariance matrix

Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex random variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$, and let $A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root of a nonnegative definite matrix $T_n$ and $X_n$ is the $n\times N$ matrix of the upper-left corner of the double array. The matrix $A_n$ can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix $T_n$, or as a multivariate $F$ matrix if $T_n$ is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of $A_n$, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if \{$X_{ij}$\} and $T_n$ are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of $A_n$ are proved to have Gaussian limits, which suggests that the eigenvector matrix of $A_n$ is nearly Haar distributed when $T_n$ is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

math.PR↗