Lower Semilinear Copulas

• Lower semilinear copulas are bivariate functions where the entire dependence structure is encoded by the diagonal section and extended via linearity on lower regions.
• They facilitate exact calculations of dependence measures such as Kendall’s τ, Spearman’s ρ, and Chatterjee’s ξ using explicit closed-form formulas.
• Their algebraic closure under the star product and convex-analytic representation provide a rigorous framework for extremal dependence and optimization of copula integrals.

Searching arXiv for recent and foundational papers on lower semilinear copulas. Lower semilinear copulas are a bivariate copula class in which the dependence structure is determined entirely by the diagonal section and extended to the unit square by linearity on lower triangular regions. In the modern formulation, a copula $C$ is lower semilinear if, for every $x\in(0,1]$, the mappings $t\mapsto C(t,x)$ and $t\mapsto C(x,t)$ are linear on $[0,x]$, and such copulas admit the closed form

$S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$

where $\delta(t)=C(t,t)$ is the copula diagonal (Maislinger et al., 2024). This class has become a tractable setting for studying convex geometry, star-product dynamics, exact regions of dependence measures, and extremal constructions. The term also appears in a broader, contextual sense in dependence optimization, where lower extremizers of copula integral problems are realized by shuffle-of-$M$ constructions supported on finitely many line segments (Hofer et al., 2014).

1. Definition and diagonal characterization

A bivariate copula is a map $C:[0,1]^2\to[0,1]$ satisfying the boundary conditions

$C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$

together with 2-increasingness: $x\in(0,1]$0 for all $x\in(0,1]$1, $x\in(0,1]$2 (Hofer et al., 2014). Its diagonal section is $x\in(0,1]$3.

Lower semilinear copulas are characterized by a specific class of diagonals. A function $x\in(0,1]$4 is admissible for the lower semilinear class if it is non-decreasing, 2-Lipschitz, satisfies $x\in(0,1]$5, and obeys the two shape constraints

$x\in(0,1]$6

on $x\in(0,1]$7 (Fuchs et al., 31 Jul 2025, Durante et al., 14 Jan 2025). The corresponding diagonal class is denoted $x\in(0,1]$8 in one formulation and $x\in(0,1]$9 in another equivalent semilinear formulation (Fuchs et al., 31 Jul 2025, Durante et al., 14 Jan 2025).

These monotonicity constraints imply

$t\mapsto C(t,x)$0

and admit the almost-everywhere differential characterization

$t\mapsto C(t,x)$1

for $t\mapsto C(t,x)$2-almost every $t\mapsto C(t,x)$3 (Fuchs et al., 31 Jul 2025, Maislinger et al., 2024). The equivalence between the ratio conditions and the differential inequality is central because it converts the geometric definition of the class into a one-dimensional regularity condition on $t\mapsto C(t,x)$4.

For every $t\mapsto C(t,x)$5, the associated copula $t\mapsto C(t,x)$6 is symmetric: $t\mapsto C(t,x)$7 and satisfies

$t\mapsto C(t,x)$8

where $t\mapsto C(t,x)$9 is the independence copula and $t\mapsto C(x,t)$0 is the upper Fréchet–Hoeffding bound (Maislinger et al., 2024). The family $t\mapsto C(x,t)$1 is convex and compact in the uniform metric, and compact under the sup norm is likewise emphasized in later work (Maislinger et al., 2024, Fuchs et al., 31 Jul 2025).

2. Geometric structure and relation to semilinear copulas

In the semilinear literature, lower semilinear copulas are treated as the operative subclass of semilinear copulas. One formulation states that a lower semilinear copula is “constructed from a linear interpolation between the values that $t\mapsto C(x,t)$2 assumes at the lower boundaries of the unit square and the values that $t\mapsto C(x,t)$3 assumes on the diagonal section,” and then gives the explicit representation

$t\mapsto C(x,t)$4

with the convention $t\mapsto C(x,t)$5 (Durante et al., 14 Jan 2025). In that treatment, upper semilinear copulas are reduced to the lower case through survival copulas, and the analysis is therefore carried out on the lower semilinear form (Durante et al., 14 Jan 2025).

This diagonal encoding makes the class effectively one-dimensional: the full bivariate copula is reconstructed from $t\mapsto C(x,t)$6 by linear formulas on the two triangles determined by the main diagonal of the unit square (Fuchs et al., 31 Jul 2025). This suggests that lower semilinear copulas occupy an intermediate position between fully parametric copula families and general nonparametric copulas: they remain infinite-dimensional, but their admissible geometry is sharply constrained by the diagonal.

Several special diagonal families recur throughout the literature. Two principal examples are

$t\mapsto C(x,t)$7

in the notation of the dependence-region analysis (Fuchs et al., 31 Jul 2025), and

$t\mapsto C(x,t)$8

in the star-product analysis (Maislinger et al., 2024). The notation differs across papers, but in each case these piecewise-defined diagonals generate structurally simple lower semilinear copulas and serve as sharp extremizers for several inequalities.

A related but distinct use of “lower semilinear” appears in the optimization of copula integrals. There, the paper does not propose an axiomatic standalone definition of lower semilinear copulas; instead, it identifies copulas attaining optimal lower bounds for piecewise constant integrands as shuffles of $t\mapsto C(x,t)$9 supported on finitely many line segments, obtained by reversing diagonal versus antidiagonal orientation in the extremal construction (Hofer et al., 2014). In that context, the phrase describes lower extremizers with piecewise linear support geometry rather than the diagonal-based family $[0,x]$0.

3. Algebraic structure and the star product

A major structural result is that the class $[0,x]$1 is closed under the star product, also called the Markov product (Maislinger et al., 2024). For copulas $[0,x]$2 and $[0,x]$3, the star product is defined by

$[0,x]$4

or equivalently in kernel form by

$[0,x]$5

(Maislinger et al., 2024). If $[0,x]$6, then $[0,x]$7 is again lower semilinear, with explicit formula

$[0,x]$8

This formula shows that closure is not merely abstract; the product can be computed directly at the diagonal level (Maislinger et al., 2024).

Indeed, the star product transfers to diagonals via

$[0,x]$9

with $S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$0, and

$S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$1

(Maislinger et al., 2024). The map $S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$2 is therefore an isomorphism between the copula class equipped with $S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$3 and the diagonal class equipped with the induced operation.

The Markov kernel of an LSL copula also admits an explicit form. If $S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$4 is a measurable version of $S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$5, then

$S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$6

The singular mass is

$S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$7

(Maislinger et al., 2024). These formulas connect the diagonal representation to transition-kernel and singular/absolutely-continuous decompositions.

The iterated star products

$S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$8

converge uniformly to a limit diagonal $S_\delta(u,v):= \begin{cases} v\,\dfrac{\delta(u)}{u}, & v\le u,\[1ex] u\,\dfrac{\delta(v)}{v}, & v>u, \end{cases}$9, and the corresponding copulas satisfy

$\delta(t)=C(t,t)$0

uniformly (Maislinger et al., 2024). The limiting copula is idempotent: $\delta(t)=C(t,t)$1 Moreover, the idempotent lower semilinear copulas are exactly the one-parameter family $\delta(t)=C(t,t)$2, where

$\delta(t)=C(t,t)$3

Geometrically, these are ordinal sums of $\delta(t)=C(t,t)$4 and $\delta(t)=C(t,t)$5 on the interval decomposition $\delta(t)=C(t,t)$6 (Maislinger et al., 2024).

4. Dependence measures and exact attainable regions

Lower semilinear copulas form a particularly tractable class for exact calculations of dependence measures (Fuchs et al., 31 Jul 2025). For $\delta(t)=C(t,t)$7, the paper gives

$\delta(t)=C(t,t)$8

and

$\delta(t)=C(t,t)$9

for Kendall’s $M$0, Spearman’s $M$1, and Spearman’s footrule $M$2, respectively (Fuchs et al., 31 Jul 2025). A different paper records the formulas

$M$3

(Maislinger et al., 2024). Since these formulas are stated differently in the two papers, they should be read as paper-specific conventions or definitions in the respective analyses rather than automatically identified without qualification.

A central result is the exact attainable $M$4-region

$M$5

together with the sharp inequalities

$M$6

for every $M$7 (Fuchs et al., 31 Jul 2025). This resolves a conjecture attributed there to Maislinger–Trutschnig. The lower boundary is attained by the family $M$8, and the upper boundary by $M$9 (Fuchs et al., 31 Jul 2025).

The same paper derives the exact $C:[0,1]^2\to[0,1]$0-region: $C:[0,1]^2\to[0,1]$1 again with both inequalities sharp. The lower equality is attained exclusively by $C:[0,1]^2\to[0,1]$2, while the upper equality is attained by $C:[0,1]^2\to[0,1]$3 (Fuchs et al., 31 Jul 2025). Combining the two regions yields the exact $C:[0,1]^2\to[0,1]$4-region

$C:[0,1]^2\to[0,1]$5

equivalently

$C:[0,1]^2\to[0,1]$6

which is also convex and compact (Fuchs et al., 31 Jul 2025).

Chatterjee’s rank correlation $C:[0,1]^2\to[0,1]$7 is treated as a directed dependence measure. For lower semilinear copulas, it admits the closed form

$C:[0,1]^2\to[0,1]$8

Because the integrand is nonnegative under the lower semilinear differential constraint, one obtains

$C:[0,1]^2\to[0,1]$9

(Fuchs et al., 31 Jul 2025). Equality holds if and only if, for $C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$0-almost every $C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$1,

$C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$2

(Fuchs et al., 31 Jul 2025).

The exact $C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$3-region is

$C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$4

so the attainable set is

$C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$5

(Fuchs et al., 31 Jul 2025). The lower bound is attained by the power diagonals $C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$6, $C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$7, for which

$C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$8

(Fuchs et al., 31 Jul 2025). The upper bound $C(x,0)=C(0,y)=0,\qquad C(x,1)=x,\qquad C(1,y)=y,$9 is sharp for the extremal diagonals $x\in(0,1]$00 and $x\in(0,1]$01.

A further consequence is that, on $x\in(0,1]$02,

$x\in(0,1]$03

all sharply (Fuchs et al., 31 Jul 2025). Equality in $x\in(0,1]$04 occurs exclusively for $x\in(0,1]$05, while equality in $x\in(0,1]$06 occurs exclusively for $x\in(0,1]$07 (Fuchs et al., 31 Jul 2025). The paper stresses that these inequalities may fail outside the genuine lower semilinear class, using Marshall–Olkin examples to show that the full copula structure matters, not only the diagonal (Fuchs et al., 31 Jul 2025).

5. Extremal geometry and convex-analytic representation

The geometry of lower semilinear copulas is particularly amenable to convex analysis because the map from diagonal to copula preserves convex combinations: $x\in(0,1]$08 (Durante et al., 14 Jan 2025). Consequently, extremality in the semilinear copula class is equivalent to extremality of the corresponding diagonal.

The principal theorem states that for an admissible diagonal $x\in(0,1]$09, the corresponding semilinear copula $x\in(0,1]$10 is an extreme point of the semilinear copula class if and only if $x\in(0,1]$11 is an extreme point of the admissible diagonal set (Durante et al., 14 Jan 2025). The extreme diagonals are characterized by the measure-theoretic condition

$x\in(0,1]$12

Thus a diagonal is extreme exactly when the normalized derivative does not take interior values between the two boundary slopes $x\in(0,1]$13 and $x\in(0,1]$14 on a set of positive measure (Durante et al., 14 Jan 2025).

This criterion shows that extremal lower semilinear copulas are those for which the diagonal saturates the admissible differential constraints almost everywhere. The proof strategy in the paper uses explicit multiplicative perturbations

$x\in(0,1]$15

to split non-extreme diagonals into two distinct admissible ones while preserving the semilinear constraints (Durante et al., 14 Jan 2025).

The compactness and convexity of the semilinear class imply a Krein–Milman representation: $x\in(0,1]$16 in the appropriate sense (Durante et al., 14 Jan 2025). The paper also invokes Choquet’s theorem to represent semilinear copulas as mixtures of extreme semilinear copulas. A distinguished extreme family is generated by

$x\in(0,1]$17

which interpolates between $x\in(0,1]$18 and $x\in(0,1]$19 in a patchwork-style construction (Durante et al., 14 Jan 2025).

For this family, the paper gives

$x\in(0,1]$20

and

$x\in(0,1]$21

For general semilinear copulas represented by a probability measure $x\in(0,1]$22, these quantities become integrals against $x\in(0,1]$23 (Durante et al., 14 Jan 2025). This supports the interpretation that dependence functionals on the full class reduce to moment-type quantities on an extreme-point parametrization.

The same convex-analytic perspective is also used to study asymmetry and radial asymmetry maps. Because semilinear copulas are exchangeable, the relevant optimization is transferred to extreme diagonals using the Bauer maximum principle (Durante et al., 14 Jan 2025). This suggests that extreme lower semilinear copulas serve not only as boundary objects in the class itself but also as maximizers of a broad family of dependence functionals.

6. Extremal constructions, regularization, and related formulations

A separate line of work connects lower semilinear behavior to extremal dependence optimization for copula integrals. For a piecewise constant function on an $x\in(0,1]$24 grid,

$x\in(0,1]$25

the maximization problem

$x\in(0,1]$26

reduces to the linear assignment problem

$x\in(0,1]$27

(Hofer et al., 2014). The maximizing copula is a shuffle of $x\in(0,1]$28 associated with an optimal assignment permutation and supported on a finite collection of line segments. The lower bound follows by applying the same theorem to $x\in(0,1]$29: $x\in(0,1]$30 (Hofer et al., 2014).

In this assignment-based setting, lower extremizers are described as lower-bound analogues of semilinear or extremal copulas: their mass is concentrated on finitely many line segments and the “lower” version is obtained by reversing diagonal versus antidiagonal orientation (Hofer et al., 2014). This is not the same as the diagonal-encoded class $x\in(0,1]$31, but it is closely related at the level of support geometry and extremal dependence structure. A plausible implication is that the phrase “lower semilinear” can carry two overlapping meanings in the literature: a specific diagonal-based copula family, and a broader descriptive label for lower extremal copulas with piecewise linear support.

For general continuous $x\in(0,1]$32, the same paper approximates extremal values by dyadic step functions $x\in(0,1]$33 and $x\in(0,1]$34, with optimizing copulas $x\in(0,1]$35 and $x\in(0,1]$36, and proves

$x\in(0,1]$37

(Hofer et al., 2014). If $x\in(0,1]$38 is Lipschitz with constant $x\in(0,1]$39, then

$x\in(0,1]$40

(Hofer et al., 2014). These results position semilinear-type extremizers as computational tools for dependence optimization.

Another related direction studies regularization of the Fréchet–Hoeffding bounds $x\in(0,1]$41 and $x\in(0,1]$42, both singular copulas whose mass is concentrated on one-dimensional sets (Björnham et al., 2016). The lower bound $x\in(0,1]$43 has all probability mass on the anti-diagonal $x\in(0,1]$44, while $x\in(0,1]$45 has all mass on the diagonal $x\in(0,1]$46 (Björnham et al., 2016). Through an explicit disk-averaging construction, the paper obtains absolutely continuous copulas $x\in(0,1]$47 and $x\in(0,1]$48 under sufficient differential conditions on a $x\in(0,1]$49 radius function $x\in(0,1]$50 (Björnham et al., 2016).

The paper does not use the term lower semilinear copulas explicitly, but it emphasizes the relevance of regularizing the lower Fréchet–Hoeffding bound for lower-bound-type dependence models (Björnham et al., 2016). The resulting $x\in(0,1]$51 preserves lower-bound geometry in an averaged form, with mass supported in a band around the anti-diagonal: $x\in(0,1]$52 (Björnham et al., 2016). This suggests a bridge between singular piecewise-linear lower extremizers and absolutely continuous copulas with similar geometric concentration. The paper also states that the method cannot prove regularity beyond $x\in(0,1]$53, because the auxiliary function $x\in(0,1]$54 used in the construction is $x\in(0,1]$55 but not $x\in(0,1]$56 (Björnham et al., 2016).

7. Scope, interpretations, and related distinctions

Several distinctions are important for interpreting the term correctly.

First, lower semilinear copulas in the diagonal-based sense form a specific bivariate class $x\in(0,1]$57 characterized by linearity on lower triangular regions and by the diagonal constraints on $x\in(0,1]$58 (Maislinger et al., 2024, Fuchs et al., 31 Jul 2025). They are symmetric and satisfy $x\in(0,1]$59 (Maislinger et al., 2024).

Second, semilinear copulas in the convex-geometric literature are effectively identified with the lower semilinear form, because upper semilinear copulas are handled through survival copulas (Durante et al., 14 Jan 2025). As a result, many “semilinear copula” theorems in that literature are, operationally, theorems about lower semilinear copulas.

Third, in dependence-optimization problems, “lower semilinear” may refer more loosely to lower extremal shuffles of $x\in(0,1]$60 supported on finitely many segments (Hofer et al., 2014). These objects share a piecewise linear support structure with lower semilinear copulas, but they are not presented there as members of the diagonal class $x\in(0,1]$61. The distinction is substantive: one theory is class-defining, the other is optimization-driven.

Fourth, the exact inequalities for $x\in(0,1]$62 are specific to genuine lower semilinear copulas. The 2025 analysis explicitly notes that they may fail outside $x\in(0,1]$63, even for copulas whose diagonals resemble lower semilinear ones, because the full copula structure matters (Fuchs et al., 31 Jul 2025). This addresses a potential misconception that the diagonal alone controls every dependence inequality once its shape resembles an admissible lower semilinear diagonal.

Within the broader copula literature, lower semilinear copulas occupy a distinct niche. They are more rigid than general bivariate copulas, but sufficiently rich to exhibit nontrivial convex geometry, nontrivial Markov-product dynamics, exact attainable regions for dependence measures, and explicit extremal families. Their one-dimensional encoding through the diagonal explains both their tractability and their limitations: they provide exact formulas and sharp structure theorems precisely because the admissible dependence patterns are tightly constrained (Maislinger et al., 2024, Fuchs et al., 31 Jul 2025, Durante et al., 14 Jan 2025).