Table 4 audit: the degrees-of-freedom limits of the t-EV family #
Table 4 prints C^{tEV}_{0,ρ}=C^{MO} and C^{tEV}_{∞,ρ}=C^{HR}_ρ. For a fixed correlation
-1<ρ<1 we prove:
- as
ν → ∞the t-EV copulas converge pointwise to independence; hence no Hüsler–Reiss copula with positive parameter is the fixed-correlation limit; - as
ν → 0+they converge pointwise to the Marshall–Olkin copula with both weights equal to1/2+arcsin(ρ)/π, which is not independence (so not a Marshall–Olkin copula with a zero weight).
theorem
Papers.AnsariRockel2024.tEV_limit_infinity
{ι : Type u_1}
{l : Filter ι}
(ν : ι → ℝ)
(hν : ∀ (i : ι), 0 < ν i)
(hνt : Filter.Tendsto ν l Filter.atTop)
(r : ℝ)
(hr : r ∈ Set.Ioo (-1) 1)
(u v : ↑unitInterval)
:
Filter.Tendsto (fun (i : ι) => (Verification.tEV (ν i) r ⋯ ⋯).cdf ![u, v]) l
(nhds ((ProbabilityTheory.Copula.independence 2).cdf ![u, v]))
theorem
Papers.AnsariRockel2024.tEV_limit_zero
{ι : Type u_1}
{l : Filter ι}
[l.IsCountablyGenerated]
(ν : ι → ℝ)
(hν : ∀ (i : ι), 0 < ν i)
(hν1 : ∀ (i : ι), ν i ≤ 1)
(hνl : Filter.Tendsto ν l (nhds 0))
(r : ℝ)
(hr : r ∈ Set.Ioo (-1) 1)
(u v : ↑unitInterval)
:
Filter.Tendsto (fun (i : ι) => (Verification.tEV (ν i) r ⋯ ⋯).cdf ![u, v]) l
(nhds
((ProbabilityTheory.Copula.marshallOlkin (Verification.tEVZeroWeightI r) (Verification.tEVZeroWeightI r)).cdf
![u, v]))
theorem
Papers.AnsariRockel2024.tEV_printed_infinity_limit_false
(r : ℝ)
(hr : r ∈ Set.Ioo (-1) 1)
(δ : ℝ)
(hδ : 0 < δ)
:
¬∀ (u v : ↑unitInterval),
Filter.Tendsto (fun (n : ℕ) => (Verification.tEV (↑n + 1) r ⋯ ⋯).cdf ![u, v]) Filter.atTop
(nhds ((Verification.huslerReissPositive δ hδ).cdf ![u, v]))
The printed ν → ∞ limit fails: for fixed ρ∈(-1,1) the limit is independence, which is
not a Hüsler–Reiss copula with positive parameter.
The ν → 0 limit is not independence (in particular not a Marshall–Olkin copula with a
zero weight): its upper tail coefficient is 1/2+arcsin(ρ)/π>0.