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Register a new userExtreme eigenvalue statistics of $m$-dependent heavy-tailed matrices
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We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter $0<alpha<4$. Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.
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We study the statistics of the largest eigenvalues of $p times p$ sample covariance matrices $Sigma_{p,n} = M_{p,n}M_{p,n}^{*}$ when the entries of the $p times n$ matrix $M_{p,n}$ are sparse and have a distribution with tail $t^{-alpha}$, $alpha>0$. On average the number of nonzero entries of $M_{p,n}$ is of order $n^{mu+1}$, $0 leq mu leq 1$. We prove that in the large $n$ limit, the largest eigenvalues are Poissonian if $alpha<2(1+mu^{{-1}})$ and converge to a constant in the case $alpha>2(1+mu^{{-1}})$. We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.
We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of $X_N$ for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on the characteristic function method in [47], local laws for the Green function of $X_N$ in [3, 46, 51] and analytic subordination properties of the free additive convolution [24, 41]. We also prove the analogous results for high-dimensional sample covariance matrices.
Consider a $p$-dimensional population ${mathbf x} inmathbb{R}^p$ with iid coordinates in the domain of attraction of a stable distribution with index $alphain (0,2)$. Since the variance of ${mathbf x}$ is infinite, the sample covariance matrix ${mathbf S}_n=n^{-1}sum_{i=1}^n {{mathbf x}_i}{mathbf x}_i$ based on a sample ${mathbf x}_1,ldots,{mathbf x}_n$ from the population is not well behaved and it is of interest to use instead the sample correlation matrix ${mathbf R}_n= {operatorname{diag}({mathbf S}_n)}^{-1/2}, {mathbf S}_n {operatorname{diag}({mathbf S}_n)}^{-1/2}$. This paper finds the limiting distributions of the eigenvalues of ${mathbf R}_n$ when both the dimension $p$ and the sample size $n$ grow to infinity such that $p/nto gamma in (0,infty)$. The family of limiting distributions ${H_{alpha,gamma}}$ is new and depends on the two parameters $alpha$ and $gamma$. The moments of $H_{alpha,gamma}$ are fully identified as sum of two contributions: the first from the classical Marv{c}enko-Pastur law and a second due to heavy tails. Moreover, the family ${H_{alpha,gamma}}$ has continuous extensions at the boundaries $alpha=2$ and $alpha=0$ leading to the Marv{c}enko-Pastur law and a modified Poisson distribution, respectively. Our proofs use the method of moments, the path-shortening algorithm developed in [18] and some novel graph counting combinatorics. As a consequence, the moments of $H_{alpha,gamma}$ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions $H_{alpha,gamma}$ is also provided for comparison with the Marv{c}enko-Pastur law.
We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$, $B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haar distributed random unitary matrix. We establish a universal Central Limit Theorem for the linear eigenvalue statistics of $H_N$ on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics, and consists of two main steps: (1) generating Ward identities using the left-translation-invariance of the Haar measure, along with a local law for the resolvent of $H_N$ and analytic subordination properties of the free additive convolution, allow us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.
We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p_n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavytailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.
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