| 1. |
- Dörr, Philip, et al.
(författare)
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Extremes of joint inversions and descents on finite Coxeter groups
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The numbers of inversions and descents of random permutations are known to be asymptotically normal. On general finite Coxeter groups, the central limit theorem (CLT) is still valid under mild conditions. The extreme values of these two statistics are attracted by the Gumbel distribution. The joint distribution of inversions and descents is a likewise interesting object, but only the CLT on symmetric groups has been established thus far. In this paper, we comprehensively extend the knowledge of the joint distribution of inversions and descents. We prove both the CLT and the extreme value attraction for the joint distribution of inversions and descents by using Hájek projections and a suitable Gaussian approximation. On the signed permutation groups, we additionally show that these results are still valid when the choice of the random signs is biased. Furthermore, we investigate the applicability of these techniques to products of classical Weyl groups.
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| 2. |
- Fleermann, Michael, et al.
(författare)
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Large sample covariance matrices of Gaussian observations with uniform correlation decay
- 2023
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Ingår i: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 162, s. 456-480
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Tidskriftsartikel (refereegranskat)abstract
- 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 an/n, where an may grow mildly to infinity. We employ the method of moments tightly: We identify the exact condition on the growth of an 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 Vn under a uniform correlation bound of C/nδ, 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.
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| 3. |
- Gusakova, Anna, et al.
(författare)
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The volume of random simplices from elliptical distributions in high dimension
- 2023
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Ingår i: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 164, s. 357-382
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Heiny, Johannes, 1989-, et al.
(författare)
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Asymptotic independence of point process and Frobenius norm of a large sample covariance matrix
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 5. |
- Parolya, Nestor, et al.
(författare)
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Logarithmic law of large random correlation matrices
- 2024
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Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 30:1, s. 346-370
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Tidskriftsartikel (refereegranskat)abstract
- Consider a random vector y=Σ1∕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 Σ1∕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.
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