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Lower tail dependence of non-strict Archimedean copulas

If the inverse generator ψ vanishes on [a, ∞) with a > 0 and the generator satisfies φ(t) → a as t → 0⁺, then the diagonal δ(t) = ψ(2 φ(t)) vanishes near zero, so the lower tail-dependence coefficient is λ_L = 0 (Nelsen, An Introduction to Copulas, second edition, Section 5.4; for non-strict generators C(t, t) = 0 for small t). This applies to the non-strict families 11, 18, 21 and 22 of Nelsen's Table 4.1.

Two further tail coefficients follow from the explicit diagonals: family 18 has upper tail-dependence coefficient λ_U = 1, and family 16 (for θ > 0) has lower tail-dependence coefficient λ_L = 1/2 (values as in Ansari and Rockel, Table 5).

theorem ProbabilityTheory.Copula.hasLowerTailDependence_zero_of_eventually_diagonal_eq_zero (C : Copula 2) (h : ∀ᶠ (t : ↑unitInterval) in nhdsWithin 0 (Set.Ioi 0), C.diagonal t = 0) : C.HasLowerTailDependence 0

A copula whose diagonal vanishes near zero has lower tail-dependence coefficient zero.

theorem ProbabilityTheory.Copula.BivariateGenerator.hasLowerTailDependence_zero (g : BivariateGenerator) {a : ℝ} (ha : 0 < a) (hzero : ∀ (s : ℝ), a ≤ s → g.toFun s = 0) (hφ : Filter.Tendsto g.invFun (nhdsWithin 0 (Set.Ioi 0)) (nhds a)) : g.copula.HasLowerTailDependence 0

A non-strict generator (ψ = 0 on [a, ∞), a > 0) whose generator tends to a at zero gives a copula without lower tail dependence.

theorem ProbabilityTheory.Copula.hasLowerTailDependence_nelsen11 (θ : ℝ) (hθ : 0 < θ) (h2 : θ ≤ 1 / 2) : (nelsen11 θ hθ h2).HasLowerTailDependence 0

Nelsen's family 11 has no lower tail dependence.

theorem ProbabilityTheory.Copula.hasLowerTailDependence_nelsen18 (θ : ℝ) (hθ : 2 ≤ θ) : (nelsen18 θ hθ).HasLowerTailDependence 0

Nelsen's family 18 has no lower tail dependence.

theorem ProbabilityTheory.Copula.hasLowerTailDependence_nelsen21 (θ : ℝ) (hθ : 1 ≤ θ) : (nelsen21 θ hθ).HasLowerTailDependence 0

Nelsen's family 21 has no lower tail dependence.

theorem ProbabilityTheory.Copula.hasLowerTailDependence_nelsen22 (θ : ℝ) (hθ : 0 < θ) (h1 : θ ≤ 1) : (nelsen22 θ hθ h1).HasLowerTailDependence 0

Nelsen's family 22 has no lower tail dependence.

theorem ProbabilityTheory.Copula.hasUpperTailDependence_nelsen18 (θ : ℝ) (hθ : 2 ≤ θ) : (nelsen18 θ hθ).HasUpperTailDependence 1

Nelsen's family 18 has upper tail-dependence coefficient one.

theorem ProbabilityTheory.Copula.hasLowerTailDependence_nelsen16 (θ : ℝ) (hθ : 0 < θ) : (nelsen16 θ ⋯).HasLowerTailDependence (1 / 2)

Nelsen's family 16 has lower tail-dependence coefficient 1/2 for every θ > 0.