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The volume of random simplices from elliptical distributions in high dimension

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  • Gusakova, Anna
  • Heiny, Johannes
  • Thäle, Christoph

Abstract

Random simplices and more general random convex bodies of dimension p in Rn 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.

Suggested Citation

  • Gusakova, Anna & Heiny, Johannes & Thäle, Christoph, 2023. "The volume of random simplices from elliptical distributions in high dimension," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 357-382.
    • Handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:357-382
    DOI: 10.1016/j.spa.2023.07.012
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    References listed on IDEAS

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    1. Heiny, Johannes & Mikosch, Thomas, 2018. "Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2779-2815.
    2. Dette, Holger & Dörnemann, Nina, 2020. "Likelihood ratio tests for many groups in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
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