| 1. |
- Fleermann, Michael, et al.
(author)
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Large sample covariance matrices of Gaussian observations with uniform correlation decay
- 2023
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In: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 162, s. 456-480
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Journal article (peer-reviewed)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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| 2. |
- Gusakova, Anna, et al.
(author)
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The volume of random simplices from elliptical distributions in high dimension
- 2023
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In: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 164, s. 357-382
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Journal article (peer-reviewed)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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