Abstract. In this paper we study the joint distributional convergence of the largest eigenvalues of the sample covariance matrix of a p-dimensional time series with iid entries when p converges to infinity together with the sample size n. We consider only heavy-tailed time series in the sense that the entries satisfy some regular variation condition which ensures that their fourth moment is infinite. In this case, Soshnikov [31,32] and Auffinger et al. [2] proved the weak convergence of the point processes of the normalized eigenvalues of the sample covariance matrix towards an inhomogeneous Poisson process which implies in turn that the largest eigenvalue converges in distribution to a Fréchet distributed random variable. They proved these results under the assumption that p and n are proportional to each other. In this paper we show that the aforementioned results remain valid if p grows at any polynomial rate. The proofs are different from those in [2, 31, 32]; we employ large deviation techniques to achieve them. The proofs reveal that only the diagonal of the sample covariance matrix is relevant for the asymptotic behavior of the largest eigenvalues and the corresponding eigenvectors which are close to the canonical basis vectors. We also discuss extensions of the results to sample autocovariance matrices.

Abstract. We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = pn 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 heavy-tailed 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 functionals 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.1. Estimation of the largest eigenvalues: an overview in the iid case 1.1. The light-tailed case. One of the exciting new areas of statistics is concerned with analyses of large data sets. For such data one often studies the dependence structure via covariances and correlations. In this paper we focus on one aspect: the estimation of the eigenvalues of the covariance matrix of a multivariate time series when the dimension p of the series increases with the sample size n. In particular, we are interested in limit theory for the largest eigenvalues of the sample covariance matrix. This theory is closely related to topics from classical extreme value theory such as maximum domains of attraction with the corresponding normalizing and centering constants for maxima; cf. Embrechts et al. [17], Resnick [29,30]. Moreover, point process convergence with limiting Poisson and cluster Poisson processes enters in a natural way when one describes the joint convergence of the largest eigenvalues of the sample covariance matrix. Large deviation techniques find applications, linking extreme value theory with random walk theory and point process convergence. The objective of this paper is to illustrate some of the main developments in random matrix theory for the particular case of the sample covariance matrix of multivariate time series with independent or dependent entries. We give special emphasis to the heavy-tailed case when extreme value theory enters in a rather straightforward way.Classical multivariate time series analysis deals with observations which assume values in a pdimensional space where p is "relatively small" compared to the sample size n. With the availability of large data sets p can be "large" relative to n. One of the possible consequences is that standard asymp...

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from n independent observations of a p-dimensional time series with iid components converge almost surely to (1+ √ γ) 2 and (1− √ γ) 2 , respectively, as n → ∞, if p/n → γ ∈ (0, 1] and the truncated variance of the entry distribution is "almost slowly varying", a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marčenko-Pastur law, which extends a result in [7]. We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study. In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.1991 Mathematics Subject Classification. Primary 60B20; Secondary 60F05 60F10 60G10 60G55 60G70.

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We study the limiting spectral distribution of sample covariance matrices XX T , where X are p × n random matrices with correlated entries and p/n → y ∈ [0, ∞). If y > 0, we obtain the Marčenko-Pastur distribution and in the case y = 0 the semicircle distribution after appropriate rescaling. The entries we consider are Curie-Weiss spins, which are correlated random signs, where the degree of the correlation is governed by an inverse temperature β > 0. The model exhibits a phase transition at β = 1. The correlation between any two entries is of order O((np) −1 ) for β ∈ (0, 1), O((np) −1/2 ) for β = 1, and for β > 1 the correlation does not vanish in the limit. In our proofs we use Stieltjes transforms and concentration of random quadratic forms.