{"ID":5675377,"CreatedAt":"2026-07-03T01:40:09.565152011Z","UpdatedAt":"2026-07-07T01:06:03.009715918Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.02169","arxiv_id":"2607.02169","title":"A note on \"The volume of random simplices from elliptical distributions in high dimension\"","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 γwithin (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.","short_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 r...","url_abs":"https://arxiv.org/abs/2607.02169","url_pdf":"https://arxiv.org/pdf/2607.02169v1","authors":"[\"Shan Xizheng\",\"Li Yanpeng\"]","published":"2026-07-02T13:39:48Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\"]","methods":"[]","has_code":false}