关于等价耦合与全变分的一些对偶结果
概率论
2023-12-06 v2
摘要
设 ( Ω , F ) (\Omega,\mathcal{F}) ( Ω , F ) P ( F ) \mathcal{P}(\mathcal{F}) P ( F ) F \mathcal{F} F E ⊂ Ω × Ω E\subset\Omega\times\Omega E ⊂ Ω × Ω E ∈ F ⊗ F E\in\mathcal{F}\otimes\mathcal{F} E ∈ F ⊗ F x ∼ y x\sim y x ∼ y ⇔ \Leftrightarrow ⇔ ( x , y ) ∈ E (x,y)\in E ( x , y ) ∈ E Ω \Omega Ω σ \sigma σ G 0 \mathcal{G}_0 G 0 G 1 \mathcal{G}_1 G 1 μ , ν ∈ P ( F ) \mu,\,\nu\in\mathcal{P}(\mathcal{F}) μ , ν ∈ P ( F ) inf P ∈ Γ ( μ , ν ) ( 1 − P ( E ) ) = \norm μ − ν G 1 and min P ∈ Γ ( μ , ν 0 ) ( 1 − P ( E ) ) = \norm μ − ν G 0 . \inf_{P\in\Gamma(\mu,\nu)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}_1}\quad\text{and}\quad\min_{P\in\Gamma(\mu,\nu_0)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}_0}. P ∈ Γ ( μ , ν ) inf ( 1 − P ( E )) = \norm μ − ν G 1 and P ∈ Γ ( μ , ν 0 ) min ( 1 − P ( E )) = \norm μ − ν G 0 . ν 0 ∈ P ( F ) \nu_0\in\mathcal{P}(\mathcal{F}) ν 0 ∈ P ( F ) G 0 \mathcal{G}_0 G 0 ν 0 = ν \nu_0=\nu ν 0 = ν G 0 ⊂ F \mathcal{G}_0\subset\mathcal{F} G 0 ⊂ F G 1 ⊂ F ^ \mathcal{G}_1\subset\widehat{\mathcal{F}} G 1 ⊂ F F ^ \widehat{\mathcal{F}} F F \mathcal{F} F σ \sigma σ μ , ν ∈ P ( F ) \mu,\,\nu\in\mathcal{P}(\mathcal{F}) μ , ν ∈ P ( F ) σ \sigma σ G ( μ , ν ) ⊂ F \mathcal{G}(\mu,\nu)\subset\mathcal{F} G ( μ , ν ) ⊂ F inf P ∈ Γ ( μ , ν ) ( 1 − P ( E ) ) = \norm μ − ν G ( μ , ν ) . \inf_{P\in\Gamma(\mu,\nu)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}(\mu,\nu)}. P ∈ Γ ( μ , ν ) inf ( 1 − P ( E )) = \norm μ − ν G ( μ , ν ) .
引用
@article{arxiv.2311.08794,
title = {Some duality results for equivalence couplings and total variation},
author = {Luca Pratelli and Pietro Rigo},
journal= {arXiv preprint arXiv:2311.08794},
year = {2023}
}
备注
11 pages