We provide asymptotic theory for the joint distribution of Xinv and Xdes, the numbers of inversions and descents of random permutations. Recently, [14] proved that Xinv, respectively, Xdes, is in the maximum domain of attraction of the Gumbel distribution. To tackle the dependency between these two permutation statistics, we use Hájek projections and a suitable quantitative Gaussian approximation. We show that (Xinv,Xdes) is in the maximum domain of attraction of the two-dimensional Gumbel distribution with independent margins. This result can be stated in the broader combinatorial framework of finite Coxeter groups, on which our method also yields the central limit theorem for (Xinv,Xdes) and various other permutation statistics as a novel contribution. In particular, signed permutation groups with random biased signs and products of classical Weyl groups are investigated.

In this paper, we consider the empirical spectral distribution of the sample correlation matrix and investigate its asymptotic behavior under mild assumptions on the data’s distribution, when dimension and sample size increase at the same rate. First, we give a characterization for the limiting spectral distribution to follow a Marčenko–Pastur law assuming that the underlying data matrix consists of i.i.d. entries. Subsequently, we provide the limiting spectral distribution of the sample correlation matrix when allowing for a dependence structure within the columns of the data matrix. In contrast to previous works, both the fourth moment of the data and the largest eigenvalue of the population correlation matrix may be unbounded, resulting in a fundamental structural difference. More precisely, the standard argument of approximating the sample correlation matrix by its sample covariance companion breaks down and novel techniques for tackling the challenging dependency structure of the sample correlation matrix are introduced.

A limit theorem for the largest interpoint distance of p independent and identically distributed points in Rn to the Gumbel distribution is proved, where the number of points p = pn tends to infinity as the dimension of the points n → ∞. The theorem holds under moment assumptions and corresponding conditions on the growth rate of p. We obtain a plethora of ancillary results such as the joint convergence of maximum and minimum interpoint dis-tances. Using the inherent sum structure of interpoint distances, our result is generalized to maxima of dependent random walks with non-decaying correlations and we also derive point process convergence. An application of the maximum interpoint distance to testing the equality of means for high-dimensional random vectors is presented. Moreover, we study the largest off-diagonal entry of a sample covariance matrix. The proofs are based on the Chen-Stein Poisson approximation method and Gaussian approximation to large deviation probabilities.

In this paper, we show the central limit theorem for the logarithmic determinant of the sample correlation matrix R constructed from the (p×n)(p×n)-dimensional data matrix X containing independent and identically distributed random entries with mean zero, variance one and infinite fourth moments. Precisely, we show that for p/n→γ∈(0,1) as n,p→∞ the logarithmic law

is still valid if the entries of the data matrix X follow a symmetric distribution with a regularly varying tail of index α∈(3,4). The latter assumptions seem to be crucial, which is justified by the simulations: if the entries of X have the infinite absolute third moment and/or their distribution is not symmetric, the logarithmic law is not valid anymore. The derived results highlight that the logarithmic determinant of the sample correlation matrix is a very stable and flexible statistic for heavy-tailed big data and open a novel way of analysis of high-dimensional random matrices with self-normalized entries.

Dans cet article, nous démontrons le théorème de la limite centrale pour le déterminant logarithmique d’une matrice de corrélation R construite à partir d’une matrice de données X de taille (p×n)(p×n) contenant des entrées indépendantes d’espérance 0, variance 1 et quatrième moment infini. Plus précisément, nous démontrons que dans le régime p/n→γ∈(0,1) quand n,p→∞ la loi logarithmique

est toujours valable si les entrées de la matrice de données X suivent une distribution symétrique avec une queue à variation régulière d’indice α∈(3,4). Ces dernières conditions semblent être cruciales, ce qui est justifié par les simulations : si les entrées de X n’ont pas de troisième moment et/ou si leur distribution n’est pas symétrique, la loi logarithmique n’est plus valable. Les résultats obtenus mettent en évidence que le déterminant logarithmique d’une matrice de corrélation est une statistique très stable et flexible pour les données massives à queue lourde et ouvrent une nouvelle voie pour analyser les grandes matrices aléatoires avec entrées auto-normalisées.

Consider a random vector y=Σ^1∕2x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ^1∕2 is a deterministic p×p matrix such that the eigenvalues of the population correlation matrix R of y are uniformly bounded away from zero and infinity. In this paper, we find that the log determinant of the sample correlation matrix based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for p∕n→γ∈(0,1] and p≤n. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of y is unknown, we show that after re-centering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. Finally, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case R=I, the test statistic becomes distribution-free and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.

The convergence of a sequence of point processes with dependent points, defined by a symmetric function of iid high-dimensional random vectors, to a Poisson random measure is proved. This also implies the convergence of the joint distribution of a fixed number of upper order statistics. As applications of the result a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices.

This paper takes a different look on the problem of testing the mutual independence of the components of a high-dimensional vector. Instead of testing if all pairwise associations (e.g., all pairwise Kendall’s τ) between the components vanish, we are interested in the (null) hypothesis that all pairwise associations do not exceed a certain threshold in absolute value. The consideration of these hypotheses is motivated by the observation that in the high-dimensional regime, it is rare, and perhaps impossible, to have a null hypothesis that can be exactly modeled by assuming that all pairwise associations are precisely equal to zero. The formulation of the null hypothesis as a composite hypothesis makes the problem of constructing tests nonstandard and in this paper we provide a solution for a broad class of dependence measures, which can be estimated by U-statistics. In particular, we develop an asymptotic and a bootstrap level α-test for the new hypotheses in the high-dimensional regime. We also prove that the new tests are minimax-optimal and investigate their finite sample properties by means of a small simulation study and a data example.

We derive the Marchenko–Pastur (MP) law for sample covariance matrices of the form , where X is a p × n data matrix and p/n → y ∈ (0,∞) as n, p → ∞. We assume the data in X stems from a correlated joint normal distribution. In particular, the correlation acts both across rows and across columns of X, and we do not assume a specific correlation structure, such as separable dependencies. Instead, we assume that correlations converge uniformly to zero at a speed of a_n/n, where a_n may grow mildly to infinity. We employ the method of moments tightly: We identify the exact condition on the growth of a_n which will guarantee that the moments of the empirical spectral distributions (ESDs) converge to the MP moments. If the condition is not met, we can construct an ensemble for which all but finitely many moments of the ESDs diverge. We also investigate the operator norm of V_n under a uniform correlation bound of C/n^δ, where C, δ > 0 are fixed, and observe a phase transition at δ = 1. In particular, convergence of the operator norm to the maximum of the support of the MP distribution can only be guaranteed if δ > 1. The analysis leads to an example for which the MP law holds almost surely, but the operator norm remains stochastic in the limit, and we provide its exact limiting distribution.

Random simplices and more general random convex bodies of dimension p in with p ≤ n are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if p → ∞ and n → ∞ in such a way that p/n → γ ∈ (0, 1), a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of p × n random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.

A joint limit theorem for the point process of the off-diagonal entries of a sample covariance matrix S, constructed from n observations of a p-dimensional random vector with iid components, and the Frobenius norm of S is proved. In particular, assuming that p and n tend to infinity we obtain a central limit theorem for the Frobenius norm in the case of finite fourth moment of the components and an infinite variance stable law in the case of infinite fourth moment. Extending a theorem of Kallenberg, we establish asymptotic independence of the point process and the Frobenius norm of S. To the best of our knowledge, this is the first result about joint convergence of a point process of dependent points and their sum in the non-Gaussian case.