A note on "The volume of random simplices from elliptical distributions in high dimension"

Shan Xizheng; Li Yanpeng

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arXiv · 2607.02169

A note on "The volume of random simplices from elliptical distributions in high dimension"

Shan Xizheng·Li Yanpeng

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Abstract

Recent work by Gusakova et al. (Stochastic Process. Appl. 164 (2023) 357-382) has shown a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies under an elliptical framework in the high dimensional regime, that is, if p and n tend to infinity in such a way that the ratio tends to \gamma within (0,1). A technical condition (Equation (2.6) of Assumption (B) therein) requires that the population matrix AA* is close in Frobenius norm to a multiple of the identity matrix, which is rather restrictive and rules out various settings for statistical application, such as spiked models and dependent structure models. In this note we offer a general relaxation of this condition, which arrives at a reasonable condition and covers numerous scenarios, as well as consequences for the volume of general random simplices and random convex bodies. In particular, our results covers the Toeplitz/AR(1) covariance structures studied by Jiang and Pham (Ann. Stat. 53 (2025) 907-928), giving a concrete application of our theorem to high-dimensional dependent covariance models.

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Shan Xizheng, Li Yanpeng. 2026-07-02. A note on "The volume of random simplices from elliptical distributions in high dimension". https://arxiv.org/abs/2607.02169

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