[Paper Review] Extreme Geometric Quantiles Under Minimal Assumptions, with a Connection to Tukey Depth

Sibsankar Singha, Marie Kratz+1 more|arXiv (Cornell University)|Mar 4, 2026

Financial Risk and Volatility Modeling0 citations

TL;DR

The paper derives moment-free upper and lower bounds for the norms of extreme geometric quantiles in R^d and reveals a link between the lower bound and Tukey halfspace depth, with implications for heavy-tailed distributions.

ABSTRACT

Geometric (also known as spatial) quantiles, introduced by Chaudhury and representing one of the three principal approaches to defining multivariate quantiles, have been well studied in the literature. In this work, we focus on the extremal behaviour of these quantiles. We establish new extremal properties, namely general lower and upper bounds for the norm of extreme geometric quantiles, free of any moment conditions. We discuss the impact of such results on the characterization of distribution behaviour. Importantly, the lower bound can be directly linked to univariate quantiles and to halfspace (Tukey) depth central regions, highlighting a novel connection between these two fundamental notions of multivariate quantiles.

Motivation & Objective

• Motivate and study extremal behavior of geometric (spatial) quantiles in high dimensions.
• Derive bounds on the growth rate of geometric quantile norms without moment conditions.
• Uncover a link between the lower bound of extreme geometric quantiles and Tukey (halfspace) depth.
• Extend existing results under minimal assumptions and relate to univariate quantiles.
• Illustrate results with numerical examples and discuss implications for heavy-tailed distributions.

Proposed method

• Define geometric quantiles via the minimization problem in (2.1) and equivalence (2.2).
• Derive a general upper bound for ||q_X(αu)|| using a probabilistic bound and a moment-based improvement (Theorems 3.1 and Corollary 3.2).
• Develop a geometry-based lower bound by relating the extreme quantile region to Tukey depth through a cone/angle argument (Theorem 3.3).
• Obtain a directional, tractable lower bound in terms of univariate projections (Theorem 3.7).
• Analyze the role of the geometric constant M_γ and its dependence on dimension and direction (Remark 3.4).
• Extend results to integrability conditions and higher-order asymptotics under finite third moment (Theorem 4.1).

(a) Bivariate Gaussian distribution with standard Gaussian margins and $\rho=70\%$

Experimental results

Research questions

• RQ1 What are the extremal growth rates of geometric quantiles in R^d under minimal assumptions?
• RQ2 Can one bound the norm of extreme geometric quantiles without finite moments, and how?
• RQ3 How is the lower bound for extreme geometric quantiles connected to Tukey halfspace depth?
• RQ4 How do directional projections influence the behavior of geometric quantiles in high dimensions?
• RQ5 What higher-order asymptotics arise for geometric quantiles when stronger moments (e.g., third moment) exist?

Key findings

• There exist general upper and lower bounds for the norm of extreme geometric quantiles that do not require moment conditions (Theorem 3.1 and Theorem 3.3).
• The lower bound is linked to Tukey (halfspace) depth, yielding a bound in terms of univariate quantiles (Theorem 3.7).
• Under finite first moment, the upper bound is tight up to constants and scales as O(1/(1−α)) (Corollary 3.2).
• For regularly varying tails, the bounds align with known MRV rates, highlighting the bounds’ sharpness in those settings (Remark 3.8 and Proposition 3.9).
• Geometric constant M_γ governs the depth-based inclusion and affects the admissible range of α (Remark 3.4).
• Under E||X||^3 < ∞, a third-order expansion for ||q_X(αu)|| is provided, revealing skewness and tail effects in the asymptotics (Theorem 4.1).

(b) Student’s $t$ marginals with $\nu_{1}=3$ , $\nu_{2}=10$ , coupled via a Gaussian copula ( $\rho=70\%$ )