Negative Simplex Dependence (NSD)

• Negative Simplex Dependence (NSD) is defined by the inequality F_S(t) ≤ ∏F_{X_i}(t), ensuring that the probability of an aggregated sum falling below a threshold does not exceed that under independence.
• NSD plays a crucial role in risk aggregation by providing a minimal structural requirement that ensures the super-additivity of Value-at-Risk (VaR) when aggregating risks.
• NSD connects probabilistic orderings, convex geometry, and supermodular ordering, bridging extremal negative dependence with concepts such as negative association and Σ-countermonotonicity.

Negative Simplex Dependence (NSD) is a form of multivariate negative dependence designed to capture the weakest structural requirement ensuring that the lower tail probability of a sum of random variables does not exceed that under independence. NSD is characterized by a simple yet robust probabilistic inequality involving the sum and marginal distributions. It holds a fundamental position in the classification of extremal negative dependence concepts, acting as a minimal requirement in applications such as risk aggregation, where additivity properties of risk measures—especially Value-at-Risk (VaR)—are of primary interest. NSD links probabilistic orderings, convex geometry of distribution spaces, and supermodular function theory, serving as a bridge between extremal negative dependence, negative association, and independence.

1. Formal Definition

Let $X_1,\ldots,X_n$ be real-valued random variables defined on a probability space, all with prescribed continuous marginal distributions $F_{X_i}$. Denote the vector $\mathbf{X} = (X_1,\dots,X_n)$, the sum $S = \sum_{i=1}^n X_i$, and the joint CDF $$F_{\mathbf{X}}(x_1,\dots,x_n) = P(X_1 \le x_1, \ldots, X_n \le x_n)$$.

$\mathbf{X}$ is said to satisfy Negative Simplex Dependence (NSD) if, for all $t \in \mathbb{R}$,

$F_S(t) \le \prod_{i=1}^n F_{X_i}(t),$

where $F_S(t) = P(S \le t)$. In the case where $X_i \ge 0$, the domain is typically restricted to $t \ge 0$ (Mohammed, 8 Dec 2025).

This inequality says that, for any threshold $t$, the probability that the aggregated sum falls below $t$ is less than or equal to the probability that all marginals simultaneously fall below $t$ when independent. NSD may be equivalently described via integration over the $n$-simplex:

$$F_S(t) = \int_{\Delta_t} dF_{\mathbf{X}}(x) \le \prod_{i=1}^n F_{X_i}(t),$$

where $$\Delta_t = \{(x_1,\dots,x_n): x_1 + \cdots + x_n \le t\}$$ (Mohammed, 8 Dec 2025). NSD is strictly weaker than negative lower-orthant dependence (NLOD), which requires the joint CDF at all points to be no greater than the product of marginals: $$F_{\mathbf{X}}(x_1,\dots,x_n)\le\prod_{i=1}^n F_{X_i}(x_i)$$.

2. Geometric and Polyhedral Structure

In the case of discrete random vectors, especially for $d$-variate Bernoulli laws with marginal means $p = (p_1, \dots, p_d)$, the set of all joint laws with fixed marginals forms a convex polytope—the Fréchet class $B_d(p)$. NSD corresponds to requiring that the law is smaller than independence in the supermodular order, i.e., $I \preceq_{\text{sm}} I^\perp$ for $I$ a Bernoulli vector and $I^\perp$ independent Bernoullis with means $p_j$ (Cossette et al., 24 Apr 2025).

The extremal NSD structures—those saturating this inequality—form a lower-dimensional face of $B_d(p)$. Specifically, for Bernoulli vectors, the extreme NSD laws are the $\Sigma$-countermonotonic (Σ-ctm) distributions: these concentrate all their mass on those $i \in \{0,1\}^d$ whose sum $i_\bullet$ is either the integer part $m = \lfloor \sum_j p_j \rfloor$ or $m+1$. The polytope $B_d^\Sigma(p)$ formed by these distributions is an antichain in the supermodular order and is weakly minimal within the Fréchet class (Cossette et al., 24 Apr 2025).

3. Key Properties and Hierarchies of Dependence

NSD establishes an ordering among dependency structures that is weaker than NLOD and negative association, but sufficiently strong to enforce critical risk-aggregation monotonicities:

• Negative Lower-Orthant Dependence (NLOD): $F_{\mathbf{X}}(x) \le \prod_i F_{X_i}(x_i)$ for all $x$; implies NSD but not conversely (NSD might fail outside the diagonal).
• Negative Association: Requires that for disjoint subsets $A,B$ and componentwise increasing functions $f,g$, $\mathrm{Cov}(f(X_A),g(X_B)) \le 0$. Not directly implied by or implying NSD (Mohammed, 8 Dec 2025).
• Co-Monotonicity (Positive Extremal): The opposite extreme; the joint CDF is $\min_i F_{X_i}(x_i)$, and $F_S(t)$ is maximized.
• $\Sigma$-Countermonotonicity: For every nontrivial partition $I$ of $\{1,\dots,d\}$, the partial sums $(\sum_{j\in I} X_j, \sum_{j\notin I} X_j)$ are bivariate countermonotonic (i.e., minimal in convex sum order).

In the Bernoulli setting, the class $B_d^\Sigma(p)$ represents precisely the set of extreme NSD laws, and is characterized by supporting distributions only on sums $m$ and $m+1$ (Cossette et al., 24 Apr 2025).

4. Consequences for Risk Aggregation and Value-at-Risk

A major application of NSD is in establishing bounds for aggregated risk measures. For random variables $X_1,\ldots,X_n \ge 0$ with continuous margins and joint law exhibiting NSD, the following holds for all $p \in (0,1)$:

$$\mathrm{VaR}_p\left(\sum_{i=1}^n X_i\right) \ge \sum_{i=1}^n \mathrm{VaR}_p(X_i),$$

where $\mathrm{VaR}_p(Z) = \inf\{z: F_Z(z) \ge p\}$ (Mohammed, 8 Dec 2025). Thus, NSD is both necessary and sufficient for strict super-additivity of VaR for all $p$: the sum’s quantile at any level never falls below the sum of marginals’ quantiles. This property sharply distinguishes NSD from both independence (exactly additive) and positive dependence (sub-additive or even strictly so).

Examples constructed using countermonotonic continuous (e.g., Pareto/reciprocal) or discrete mutually exclusive distributions provide concrete illustrations (Mohammed, 8 Dec 2025).

5. Connections with Extremal Negative Dependence Structures

NSD is intimately related to several classical and modern negative-dependence concepts:

• Countermonotonicity: The classical bivariate case (d=2); equivalent to NSD along the diagonal.
• Pairwise Countermonotonicity: Demands all pairs are countermonotonic; rarely possible in $d>2$ except with restrictive marginals.
• Joint Mixability: All mass concentrated on constant-sum configurations; a special sharper case subsumed by NSD.
• $\Sigma$-Countermonotonicity: Generalizes countermonotonicity to $d>2$; always exists, coincides with pairwise countermonotonicity and joint mixability whenever those are possible (Puccetti et al., 2015).

Extremal NSD laws correspond to the smallest possible (in convex order) aggregate sums, and provide tight lower bounds in Fréchet classes with fixed marginals.

6. Strongly Rayleigh Property and Entropic Structure

Within the NSD/Σ-ctm face of the Bernoulli Fréchet polytope, certain distributions satisfy the strongly Rayleigh property: their multivariate generating polynomial is real stable, implying negative association and, hence, NSD (Cossette et al., 24 Apr 2025). The unique maximum-entropy point in $B_d^\Sigma(p)$ is constructed by conditioning independent Bernoulli random variables on the sum being $m$ or $m+1$. This conditional law is both strongly Rayleigh and maximally random at fixed marginals, providing an explicit representative of the strongest negative dependence compatible with NSD.

A canonical mixture (convex combination) of this strongly Rayleigh law and the independence law yields a one-parameter family of distributions interpolating monotonically in the supermodular order from extremal negative dependence to independence.

7. Summary and Comparative Table

Negative Simplex Dependence (NSD) offers a natural, minimal, and geometrically tractable framework for quantifying negative dependence in multivariate aggregation problems, especially in risk aggregation and optimization under sum constraints. The table below summarizes NSD relative to related concepts:

Concept	Definition/Characterization	Existence/Extremality
NSD	$F_S(t) \le \prod_i F_{X_i}(t)$	Always exists (weakest)
$\Sigma$-countermonotonicity	All partitions’ sums countermonotonic	Always exists; extremal NSD face
Negative Association	All increasing set functions negatively correlated	Not directly comparable
Co-monotonicity	$F_{\mathbf{X}}(x)=\min_i F_{X_i}(x_i)$	Always exists (positive extremal)

Each framework fulfills a sharply distinguished role in both theoretical and applied contexts, delineating the principal boundary cases for sum distributions under fixed marginals (Puccetti et al., 2015, Cossette et al., 24 Apr 2025, Mohammed, 8 Dec 2025).