arXiv subjects

Johannes Heiny

Publications and source records attributed to Johannes Heiny.

28 records · Page 2Linked to original sources

arXiv2021

Thin-shell theory for rotationally invariant random simplices

For fixed functions $G,H:[0,\infty)\to[0,\infty)$, consider the rotationally invariant probability density on $\mathbb{R}^n$ of the form \[ μ^n(ds) = \frac{1}{Z_n} G(\|s\|_2)\, e^{ - n H( \|s\|_2)} ds. \] We show that when $n$ is large, the Euclidean norm $\|Y^n\|_2$ of a random vector $Y^n$ distributed according to $μ^n$ satisfies a Gaussian thin-shell property: the distribution of $\|Y^n\|_2$ concentrates around a certain value $s_0$, and the fluctuations of $\|Y^n\|_2$ are approximately Gaussian with the order $1/\sqrt{n}$. We apply this thin shell property to the study of rotationally invariant random simplices, simplices whose vertices consist of the origin as well as independent random vectors $Y_1^n,\ldots,Y_p^n$ distributed according to $μ^n$. We show that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior, providing a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat. 16, 141--177 (2019)]. Finally, by relating the volumes of random simplices to random determinants, we show that if $A^n$ is an $n \times n$ random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants $c_0,c_1\in(0,\infty)$ and an absolute constant $C\in(0,\infty)$ such that \[\sup_{ s \in \mathbb{R}} \left| \mathbb{P} \left[ \frac{ \log \mathrm{det}(A^n) - \log(n-1)! - c_0 }{ \sqrt{ \frac{1}{2} \log n + c_1 }} < s \right] - \int_{-\infty}^s \frac{e^{ - u^2/2} du}{ \sqrt{ 2 π}} \right| < \frac{C}{\log^{3/2}n}, \] sharpening the $1/\log^{1/3 + o(1)}n$ bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42 (1) (2014), 146--167].

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arXiv2021

Extreme eigenvalue statistics of $m$-dependent heavy-tailed matrices

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 $0<α<4$. 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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arXiv2020

Sequential change point detection in high dimensional time series

Change point detection in high dimensional data has found considerable interest in recent years. Most of the literature either designs methodology for a retrospective analysis, where the whole sample is already available when the statistical inference begins, or considers online detection schemes controlling the average time until a false alarm. This paper takes a different point of view and develops monitoring schemes for the online scenario, where high dimensional data arrives successively and the goal is to detect changes as fast as possible controlling at the same time the probability of a type I error of a false alarm. We develop a sequential procedure capable of detecting changes in the mean vector of a successively observed high dimensional time series with spatial and temporal dependence. The statistical properties of the method are analyzed in the case where both, thesample size and dimension tend to infinity. In this scenario, it is shown that the new monitoring scheme has asymptotic level alpha under the null hypothesis of no change and is consistent under the alternative of a change in at least one component of the high dimensional mean vector. The approach is based on a new type of monitoring scheme for one-dimensional data which turns out to be often more powerful than the usually used CUSUM and Page-CUSUM methods, and the component-wise statistics are aggregated by the maximum statistic. For the analysis of the asymptotic properties of our monitoring scheme we prove that the range of a Brownian motion on a given interval is in the domain of attraction of the Gumbel distribution, which is a result of independent interest in extreme value theory. The finite sample properties of the new methodology are illustrated by means of a simulation study and in the analysis of a data example.

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arXiv2020

Point process convergence for the off-diagonal entries of sample covariance matrices

We study point process convergence for sequences of iid random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a $(2+δ)$th moment. We make heavily use of precise large deviation results for sums of iid random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.

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arXiv2020

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

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 $(1+\sqrtγ)^2$ and $(1-\sqrtγ)^2$, respectively, as $n \to \infty$, if $p/n\to γ\in (0,1]$ 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 Marchenko-Pastur law, which extends a result in Bai and Zhou (2008). 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.

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arXiv2020

Large sample autocovariance matrices of linear processes with heavy tails

We provide asymptotic theory for certain functions of the sample autocovariance matrices of a high-dimensional time series with infinite fourth moment. The time series exhibits linear dependence across the coordinates and through time. Assuming that the dimension increases with the sample size, we provide theory for the eigenvectors of the sample autocovariance matrices and find explicit approximations of a simple structure, whose finite sample quality is illustrated for simulated data. We also obtain the limits of the normalized eigenvalues of functions of the sample autocovariance matrices in terms of cluster Poisson point processes. In turn, we derive the distributional limits of the largest eigenvalues and functionals acting on them. In our proofs, we use large deviation techniques for heavy-tailed processes, point process techniques motivated by extreme value theory, and related continuous mapping arguments.

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arXiv2020

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

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.

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arXiv2019

High-dimensional sample covariance matrices with Curie-Weiss entries

We study the limiting spectral distribution of sample covariance matrices $XX^T$, where $X$ are $p\times n$ random matrices with correlated entries, for the cases $p/n\to y\in [0,\infty)$. 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 decays at a rate of $O(np)$ for $β\in (0,1)$, $O(\sqrt{np}$) 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.

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arXiv2016

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

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.

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arXiv2016

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

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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