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

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.