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Mixtures and Schur order

Mixtures preserve a common Schur upper bound. In particular, adding an independent component reduces predictability in the directional Schur order.

theorem ProbabilityTheory.Copula.integral_convex_conditionalCDF_mix_le (C D : Copula 2) (a v : ↑unitInterval) {φ : ℝ → ℝ} (hc : Continuous φ) (hv : ConvexOn ℝ (Set.Icc 0 1) φ) : ∫ (u : ↑unitInterval), φ ((C.mix D a).conditionalCDF u v) ≤ (↑a * ∫ (u : ↑unitInterval), φ (C.conditionalCDF u v)) + (1 - ↑a) * ∫ (u : ↑unitInterval), φ (D.conditionalCDF u v)

theorem ProbabilityTheory.Copula.SchurLE.mix {C D E : Copula 2} (hC : C.SchurLE E) (hD : D.SchurLE E) (a : ↑unitInterval) : (C.mix D a).SchurLE E

theorem ProbabilityTheory.Copula.schurLE_mix_independence (C : Copula 2) (a : ↑unitInterval) : (C.mix (independence 2) a).SchurLE C

theorem ProbabilityTheory.Copula.schurLE_mix_independence_mono (C : Copula 2) {a b : ↑unitInterval} (hab : a ≤ b) : (C.mix (independence 2) a).SchurLE (C.mix (independence 2) b)

Retaining a larger weight on a fixed copula increases predictability.