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.

• The paper shows that the central limit theorem for logarithmic volumes persists under significantly relaxed spectral conditions.
• It employs rigorous combinatorial and martingale techniques to analyze high-dimensional projection matrices and non-isotropic covariance models.
• The extended results encompass block-spiked, Toeplitz/AR(1), and weakly equicorrelated models, impacting high-dimensional statistical inference.

Central Limit Theorem for Logarithmic Volumes of Random Simplices: Relaxed Conditions for Elliptical Distributions

Context and Significance

The study of geometric and spectral properties of random simplices generated by high-dimensional data is fundamental in stochastic geometry and random matrix theory, with important connections to high-dimensional statistics. Recent advances, particularly the work of Gusakova et al., have established central and stable limit theorems (CLTs) for the logarithmic volumes of random simplices generated from points drawn from general elliptical distributions under the regime where both the ambient dimension $n$ and the number of points $p$ diverge such that $p/n \to \gamma \in (0,1)$ ["The volume of random simplices from elliptical distributions in high dimension", Stochastic Process. Appl. 164, 2023]. However, the enforceability of their results rested on a strong technical constraint: the population covariance matrix (of the form $\mathbf{A}\mathbf{A}^{\top}$) was required to be asymptotically close (in Frobenius norm) to a scalar multiple of the identity. This condition, while facilitating analysis, significantly limits the applicability of their CLT to a narrow set of nearly isotropic models, and excludes spiked covariance models and structured dependencies such as AR(1) or Toeplitz-type covariance, which are prevalent in high-dimensional inference.

This note addresses a key theoretical bottleneck by relaxing the aforementioned spectral constraint and establishing sufficient conditions under which the CLT for the log-volume statistic persists for a much broader class of covariance structures. The authors rigorously show that their new conditions are fulfilled by a large family of models relevant in high-dimensional statistics, including block-spiked models, banded dependence (e.g., AR(1)), and weakly equicorrelated structures.

Model Formulation and Mathematical Structure

Consider i.i.d. random vectors $$\mathbf{x}_i = \xi_i \mathbf{A} \mathbf{u}_i \in \mathbb{R}^n$$, $i=1,\ldots,p$, where $\xi_i$ are independent non-negative random radii, $\mathbf{A} \in \mathbb{R}^{n \times n}$ is deterministic full-rank, and the $\mathbf{u}_i$ are i.i.d. uniformly distributed on the unit sphere. The $p$-volume of their pinned convex hull (the simplex $p$0 with vertices $p$1) is given by

$p$2

As in earlier work, analysis focuses on the asymptotic normality of $p$3 as $p$4 ($p$5), decomposing it into a sum involving i.i.d. radial contributions and the log-determinant of a projected matrix:

$p$6

with $p$7 splitting into the sum of two independent terms due to factorization into the product of random radii and a random direction-induced matrix.

Relaxed Spectral Assumptions

The main technical contribution is a substantial weakening of the spectral condition imposed on the covariance structure. Instead of requiring

$p$8

they introduce the less restrictive condition requiring the centered and cross-deviations of eigenvalues of $p$9 to satisfy

$p/n \to \gamma \in (0,1)$0

where $p/n \to \gamma \in (0,1)$1. Boundedness of the spectral width is required but isotropy is no longer enforced.

This new spectral regime explicitly includes:

• Block and multi-spiked models, possibly with a growing number of spikes with bounded magnitude and small rank;
• Toeplitz/AR(1) autocorrelation structures with decay parameter $p/n \to \gamma \in (0,1)$2;
• Equicorrelation/interclass models (e.g., covariance of the form $p/n \to \gamma \in (0,1)$3 with $p/n \to \gamma \in (0,1)$4).

Notably, these cases are excluded by the prior, stronger requirement and are critical in many statistical hypothesis testing and estimation setups in high-dimensional inference.

Main Theorem: CLT for Log-Determinant with Relaxed Conditions

The central result is a CLT for the logarithm of the determinant of $p/n \to \gamma \in (0,1)$5 (with $p/n \to \gamma \in (0,1)$6 constructed from the projected directions), under these relaxed eigenvalue constraints. Formally, under the high-dimensional scaling $p/n \to \gamma \in (0,1)$7 and the generalized spectral assumptions,

$p/n \to \gamma \in (0,1)$8

where the explicit centering ($p/n \to \gamma \in (0,1)$9) and scaling ($\mathbf{A}\mathbf{A}^{\top}$0) involve traces and moments of the population spectrum and projective matrix statistics.

The proof is a refinement of the approach used by Gusakova et al., employing detailed combinatorial and martingale arguments for the analysis of high-dimensional projection matrices, and building upon concentration inequalities and moment formulae for (inverse) Wishart matrices. The authors establish that fluctuations of the normalized traces of certain projection matrices are sufficiently controlled under their new conditions—indeed, the upper bound for the variance term is optimal even in the identity covariance case, further justifying the sharpness of their result.

Examples and Applications

The paper provides several concrete illustrations where the extended conditions are fulfilled:

• In multi-spiked models, $\mathbf{A}\mathbf{A}^{\top}$1 with $\mathbf{A}\mathbf{A}^{\top}$2 and bounded operator norm, the CLT applies.
• For Toeplitz or AR(1) model with covariance entries $\mathbf{A}\mathbf{A}^{\top}$3 and $\mathbf{A}\mathbf{A}^{\top}$4, the condition is satisfied.
• Weak equicorrelation cases ($\mathbf{A}\mathbf{A}^{\top}$5), relevant for inference in correlated Gaussian fields and multiple testing, are included—bypassing the limitations of earlier results.

Therefore, this extension enables rigorous statistical analysis (such as likelihood ratio tests in the spiked model and dependent covariance structures) previously unreachable in the high-dimensional limit.

Implications and Future Directions

The relaxation of spectral conditions substantially broadens the set of practically relevant models for which asymptotic normality of log-volumes (and related spectral statistics) can be established in high-dimensional geometric probability and statistics. This work has immediate ramifications for the theory of random geometric structures, the design and analysis of high-dimensional statistical tests (especially those involving structured or correlated covariance), and the understanding of the limiting spectral behavior of sample covariance-type matrices well beyond the i.i.d. or isotropic paradigm.

The technical approach—martingale decompositions, moment recursions for inverse Wisharts, and trace concentration—also offers a general framework for analyzing projections and invariants of random matrices with non-trivial limiting spectrum. Future research may address further weakening of spectral constraints and extensions to non-elliptical or heavy-tailed settings, more refined non-Gaussian universality, and implications for the extremal statistics in random polytopes or other geometric functionals.

Conclusion

By significantly weakening prior spectral constraints, this work proves a central limit theorem for the logarithm of the volume of random high-dimensional simplices generated from elliptical distributions, encompassing a wide array of structured covariance models excluded from previous analyses. The result is theoretically sharp and practically essential for both random geometric analysis and high-dimensional statistics. The methodology paves the way for further generalizations, enlarging the intersection of random matrix theory, multivariate statistics, and stochastic geometry.

Reference: "A note on 'The volume of random simplices from elliptical distributions in high dimension'" (2607.02169)