Simplex Dominance in High-Dimensions

• Simplex Dominance (SD) is a property where collapsing any point in the positive orthant onto the main diagonal does not increase a given functional’s value, ensuring key ordering in risk measures.
• SD underpins analysis of dependence structures and record statistics in multidimensional stochastic models, offering exact formulae and asymptotic insights for multivariate records.
• The synergy between SD and Negative Simplex Dependence (NSD) provides a rigorous foundation for VaR super-additivity, operating under minimal continuity assumptions.

Simplex Dominance (SD) encompasses two distinct but foundational concepts in high-dimensional probability and risk theory: (i) a functional margin-dependent ordering central to super-additivity phenomena for Value-at-Risk aggregation, and (ii) a dominance-based partial order underpinning the enumeration of multivariate records in random samples from the standard simplex. These concepts furnish a unified lens for investigating dependence structures, record statistics, and extremal behaviors in multidimensional stochastic models, with exact formulae and sharp existence results.

1. Definition and Formal Properties of Simplex Dominance

Let $n\in\mathbb{N}$ and $\Phi: [0, \infty)^n \to (-\infty, 0]$ be a functional. Define $s = \sum_{i=1}^n x_i$ for $(x_1, \dots, x_n)\in[0, \infty)^n$. The functional $\Phi$ is called simplex dominant (SD) if

$$\Phi(x_1, \dots, x_n) \ge \Phi(s, \dots, s)\quad \forall (x_1, \dots, x_n)\in [0, \infty)^n.$$

This property asserts that “collapsing” any point in the positive orthant onto the main diagonal (where all coordinates are equal to their sum) cannot increase the value of $\Phi$ (Mohammed, 8 Dec 2025).

In multivariate records contexts, the dominance order is defined for $p, q \in \mathbb{R}^d$ by $p \succ q$ if $p_i > q_i$ for all $i$. For independent samples from the simplex $$S_d = \{x\in \mathbb{R}^d : x_i\ge0,\, \sum x_i \le1\}$$, this induces combinatorial structures based on record events and orderings (Hwang et al., 2010).

2. SD in Value-at-Risk Aggregation: Theoretical Framework

Simplex Dominance plays a crucial role in determining the super-additivity of Value-at-Risk (VaR) for sums of non-negative (one-sided) random variables. Super-additivity of VaR, i.e., $_p[S] \ge \sum_{i=1}^n\,_p[X_i]$ for all $p\in(0,1),$ is characterized by the interaction of SD with Negative Simplex Dependence (NSD):

• NSD is a distributional property: for $S = \sum_{i=1}^n X_i$, $F_S(t) \le \prod_{i=1}^n F_{X_i}(t)$ for all $t\ge0$.
• SD is a property of the marginal functionals: for $\Phi(x) = \sum_{i=1}^n x_i \log F_{X_i}(x_i)$, SD must hold.

The central theorem states that if $\mathbf{X}=(X_1,\dots,X_n)$ is NSD with continuous $F_{X_i}$, and $\Phi$ is SD, then VaR is super-additive at all $p\in(0,1)$. No integrability or identical margin assumptions are required beyond continuity (Mohammed, 8 Dec 2025).

3. SD–NSD Synergy: Functional Chains and Proof Outline

The interplay between NSD and SD enables the derivation of a chain of inequalities ensuring VaR super-additivity: $$\sum_i x_i(p) \log F_{X_i}(x_i(p)) \ge s(p) \sum_i \log F_{X_i}(s(p)) \implies p \ge \prod_i F_{X_i}(s(p)) \ge F_S(s(p)),$$ where $x_i(p) =\,_p[X_i]$ and $s(p) = \sum_i x_i(p)$. The argument follows by observing that at VaR quantiles, $F_{X_i}(x_i(p))=p$; SD yields the left inequality; exponentiation and division establish the middle inequality; NSD achieves the final step. This chain yields $_p[S] \ge \sum_i\,_p[X_i]$ (Mohammed, 8 Dec 2025).

4. Examples, Counterexamples, and Impossibility Results

When SD holds (for example, with independent or specifically constructed marginal distributions such as $\mathbf{X}=(X, X, 1/X)$ with $X\sim\text{Pareto II}(1,1)$), VaR super-additivity is global. In contrast, failure of SD, as in $\mathbf{X}=(X, 1/(1+X))$ with $X\sim\text{Pareto II}(1,1)$, produces intervals where VaR is strictly sub-additive. A sharp impossibility result emerges: for non-negative random variables with all finite lower endpoints, strict VaR sub-additivity is impossible—only exact additivity under comonotonicity occurs. This logic extends to supports with finite upper endpoints by dual arguments (Mohammed, 8 Dec 2025).

Example	SD holds?	VaR Behavior
$(X, X, 1/X)$, $X\sim\text{Pareto II}$	Yes	Super-additive for all $p$
$(X, 1/(1+X))$, $X\sim\text{Pareto II}$	No	Sub-/super-additive by $p$

5. SD in Multivariate Records for Simplex Samples

In random geometric combinatorics, SD arises via the dominance order in the enumeration of multivariate records in independent samples from $S_d$:

• Pareto (nondominated) records: $p_k$ is a record if not dominated by any earlier point.
• Dominating (strong) records: $p_k$ dominates all previous $p_i$.
• Chain records: A recursively constructed increasing chain under dominance.

For samples from $S_d$, formulas for the expectation and variance of record counts are explicit. For example, the expected number of Pareto records is

$$E[X_n] = n^{1-1/d} \sum_{j=0}^{d-2} \binom{d-1}{j}(-1)^j \frac{\Gamma(d-1-j)}{j+1} + (-1)^{d-1}(\log n + \gamma) + o(1).$$

Variance constants admit multivariate integral and hypergeometric representations, and efficient Mellin-transform-based methods are available for numerical evaluation (Hwang et al., 2010).

6. Limit Laws, Asymptotics, and High-Dimensional Regimes

Central limit theorems with explicit rates are established for Pareto and chain record counts on cube and simplex models:

• Pareto/chain records converge to normal limits with Berry–Esseen rates $O((\log n)^{-1/2})$ (chain) and $O((\log n)^{-d/2})$ (Pareto).
• For dominating records on the simplex, the expectation and variance remain bounded, yielding no nondegenerate Gaussian limit.
• Leading order: Pareto records grow as $n^{1-1/d}$, chain records scale as $(1/d)\log n$, and dominating records remain $O(1)$. For large $d$, record statistics exhibit sharp asymptotic transitions (Hwang et al., 2010).

Record type	Asymptotic mean (simplex)	Variance order
Pareto	$\sim n^{1-1/d}$	$\sim n^{1-1/d}$
Chain	$\sim (1/d)\log n$	$\sim (1/d^2)\log n$
Dominating	$\to \sum_{k\ge1}(d!)^k/(dk)!$	$\to$ constant

7. Broader Implications and Extensions

The SD–NSD framework generalizes to random vectors with shifted or reflected supports by appropriate modifications to the functionals and conditions (e.g., $\mathbf{X}^\alpha = \alpha + \mathbf{X}$, $\mathbf{X}^\beta = \beta - \mathbf{X}$) and dualizes for sub-additive VaR regimes. These results impose sharp boundaries: strict VaR super-additivity (or sub-additivity) is excluded when all variables share finite upper (or lower) endpoints; only degeneracy persists. The simplex dominance order and associated record enumeration theory extend to related stochastic geometry and combinatorics contexts, providing precise technical language and quantitative foundation for high-dimensional extremes (Mohammed, 8 Dec 2025, Hwang et al., 2010).