Guangming Pan

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 <∞, and let An = (1 N T 1/2 n X n X* n (T 1/2 n , where T 1/2 n is the square root of a nonnegative definite matrix T n and X n is the n x N matrix of the upper-left comer of the double array. The matrix An 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 An, 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 An are proved to have Gaussian limits, which suggests that the eigenvector matrix of An is nearly Haar distributed when T n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

Circular law, extreme singular values and potential theory

Consider the empirical spectral distribution of complex random nxn matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.

On the Relationship Between MMSE-SIC and BI-GDFE Receivers for Large Multiple-Input Multiple-Output Channels

A minimum mean-square error (MMSE)-based iterative soft interference cancellation (MMSE-SIC) receiver has been proposed to mitigate the interferences of the multiple-input multiple-output (MIMO) channels, with reduced complexity as compared to maximum-likelihood (ML) detection. On the other hand, the block-iterative generalized decision-feedback equalizer (BI-GDFE) attains close to the performance of the MMSE-SIC receivers with further reduced complexity. The BI-GDFE, however, requires an accurate estimate of the input-decision correlation (IDC), which is a statistical reliability metric of earlier-made decisions. To date, the BI-GDFE receiver is applicable only to phase-shift-keying (PSK) modulations due to the absence of a method to estimate the IDC for higher order quadrature amplitude modulations (QAMs). In this paper, we establish the relationship between the MMSE-SIC and BI-GDFE receivers and propose an algorithm to determine the IDC for BI-GDFE from the unconditional MMSE-SIC (U-MMSE-SIC). We further analyze and compare the asymptotic performances of the two receivers for large random MIMO channels and prove that for the limiting case, the output signal-to-interference-plus-noise ratios (SINRs) at each iteration for both receivers converge in probability to their respective deterministic limits. Our simulation results have shown that the bit error rate (BER) performance of the BI-GDFE receiver with the proposed IDC selection method achieves close to that of the U-MMSE-SIC receiver with similar convergence behavior and reaches the single-user matched filter bound (MFB) with several iterations for high enough signal-to-noise ratio (SNR).

A Deterministic Equivalent for the Analysis of Non-Gaussian Correlated MIMO Multiple Access Channels

Using large-dimensional random matrix theory (RMT), we conduct mutual information analysis of a multiple-input multiple-output (MIMO) multiple access channel (MAC). Our channel model reflects the characteristics in small-cell networks where antenna correlations, line-of-sight components, and general type of fading distributions have to be included. The mutual information expression can be expressed as functionals of the Stieltjes transform through the so-called Shannon transform. Ideally, if the Stieltjes transform is known in the context of the large-dimensional RMT, then the problem is solved. However, it is difficult to derive the Stieltjes transform of the considered channel models directly, especially when the transmit correlation matrices are generally nonnegative definite and the channel entries are non-Gaussian. To overcome this, we use the generalized Lindeberg principle to show that the Stieltjes transforms of this class of random matrices with Gaussian or non-Gaussian independent entries coincide in the large-dimensional regime. This result permits to derive the deterministic equivalents (e.g., the Stieltjes transform and the ergodic mutual information) for non-Gaussian MIMO channels from the known results developed for Gaussian MIMO channels. As an application, we determine the capacity-achieving input covariance matrices for the MIMO-MACs and prove that the capacity-achieving input covariance matrices are asymptotically independent of the fading distribution.

Nonparametric estimate of spectral density functions of sample covariance matrices: A first step

The density function of the limiting spectral distribution of general sample covariance matrices is usually unknown. We propose to use kernel estimators which are proved to be consistent. A simulation study is also conducted to show the performance of the estimators.

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

On singular value distribution of large-dimensional autocovariance matrices

\mathbf{X}_p=(\mathbf{s}_1,...,\mathbf{s}_n)=(X_{ij})_{p \times n}

Convergence of the empirical spectral distribution function of Beta matrices

where

SPECTRAL STATISTICS OF LARGE DIMENSIONAL SPEARMAN'S RANK CORRELATION MATRIX AND ITS APPLICATION

X_{ij}

Testing Independence Among a Large Number of High-Dimensional Random Vectors

s are independent and identically distributed (i.i.d.) random variables with