Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: The iid case

Heiny, Johannes; Mikosch, Thomas

doi:10.1016/j.spa.2016.10.006

Cited by 24 publications

(39 citation statements)

References 26 publications

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“…The proof will be given in Section 6. We notice that this result is the same as for the iid field ((E[σ α ]) 1/α Z it ); see [21,Theorem 3.10 and Lemma 3.8]. This means that dependence within the light-tailed σ-field influences the limiting point process only through a multiplicative factor.…”

Section: )supporting

confidence: 58%

“…In this case, (1.4) and (1.6) imply that the diagonal entries (S i ) of S dominate all off-diagonal elements S ij in the sense that the asymptotic behavior of the eigenvalues of S is completely determined by the diagonal diag(S) of S. This phenomenon is described in Theorem 2.1. It is well known in the iid case when p = p n → ∞ (see [13,15,21]). Pioneering work for the largest eigenvalue of S under a more restrictive growth condition on p and α ∈ (0, 2) is due to Soshnikov [39,40] and Auffinger et al [2].…”

Section: The Stochastic Volatility Modelmentioning

confidence: 99%

“…Fix j ≥ 1 and let (k) be the integer sequence from condition (N B). We will follow the lines of the proof of Theorem 3.11 in [21].…”

Section: )mentioning

confidence: 99%

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The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails

Heiny¹,

Mikosch²

2019

Bernoulli

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We consider a p-dimensional time series where the dimension p increases with the sample size n. The resulting data matrix X follows a stochastic volatility model: each entry consists of a positive random volatility term multiplied by an independent noise term. The volatility multipliers introduce dependence in each row and across the rows. We study the asymptotic behavior of the eigenvalues and eigenvectors of the sample covariance matrix XX ′ under a regular variation assumption on the noise. In particular, we prove Poisson convergence for the point process of the centered and normalized eigenvalues and derive limit theory for functionals acting on them, such as the trace. We prove related results for stochastic volatility models with additional linear dependence structure and for stochastic volatility models where the time-varying volatility terms are extinguished with high probability when n increases. We provide explicit approximations of the eigenvectors which are of a strikingly simple structure. The main tools for proving these results are large deviation theorems for heavy-tailed time series, advocating a unified approach to the study of the eigenstructure of heavy-tailed random matrices.MSC 2010 subject classifications: Primary 60B20; secondary 60F05 60F10 60G10 60G55 60G70.

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Section: )supporting

confidence: 58%

Section: The Stochastic Volatility Modelmentioning

confidence: 99%

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The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails

Heiny¹,

Mikosch²

2019

Bernoulli

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“…By [4,25,15], the properly normalized λ (1) converges to a Fréchet distributed random variable. The correct normalization is roughly n 4/2.1 and hence it is expected that λ (1) /n is separated from the bulk, whose behavior ultimately determines the limiting spectral distribution, which is the Marčenko-Pastur law with parameter γ = 0.1.…”

Section: Resultsmentioning

confidence: 99%

“…The case (X it ) iid and E[X 4 ] = ∞. Asymptotic theory for the eigenvalues of XX in the case of an entry distribution with infinite fourth moment was studied in [33,34,4] in the cases when p/n → γ ∈ (0, ∞), while the authors of [15,25] allowed nearly arbitrary growth of the dimension p. In their model, the entries of X are regularly varying with index α > 0, implying that…”

Section: 2mentioning

confidence: 99%

Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

Heiny

Mikosch

2018

Stochastic Processes and their Applications

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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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Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

et al. 2016

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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...

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Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: The iid case

Cited by 24 publications

References 26 publications

The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails

The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails

Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

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