论文标题
对等效耦合和总变化的强大二元性原理
A Strong Duality Principle for Equivalence Couplings and Total Variation
论文作者
论文摘要
我们介绍并研究了概率理论中通常出现的两个优化问题的双重性概念。也就是说，在一个抽象的可测量空间$（ω，\ Mathcal {f}）$上，我们考虑对$（e，\ nathcal {g}）$，其中$ e $是$ω$和$ω$和$ \ nathcal {g} $的等价关系，是一个sub-$ n us-$σ$ -algebra of $ $ $ \ nathcal} g};我们说$（e，\ nathcal {f}）$满足“强双重性”，如果$ e $是$（\ nathcal {f} \ otimes \ Mathcal \ Mathcal {f}）$ - 可衡量的，并且对于All-Oblesity Matues $ \ Mathbb {p}，\ Mathbb {p}，\ Mathbb {p} $ we $ we（y Math） $$ \ max_ {a \ in \ Mathcal {g}} \ vert \ Mathbb {p}（a） - \ Mathbb {p}'（a）\ vert = \ min _ {\ tilde {\ Mathbb {p}} \inπ（\ Mathbb {p}，\ Mathbb {p}'）}（1- \ tilde {\ tilde {\ mathbb {p}}}（e）（e）），$ wher $ \ mathbb {p} $和$ \ mathbb {p}'$的耦合空间，以及“ max”和“ min”的声明，实际上实现了上述和immimum的最高限度。本文的结果给出了足够的条件，可以使二元性保持强大，从而将Kantorovich二元性的形式扩展到一类成本函数，从拓扑的角度来看，这些函数是不规则的，但从描述性集合理论的角度进行了规则。给定条件恢复或增强了经典结果，并且在随机演算，点过程理论和随机序列模拟中产生了新的后果。
We introduce and study a notion of duality for two classes of optimization problems commonly occurring in probability theory. That is, on an abstract measurable space $(Ω,\mathcal{F})$, we consider pairs $(E,\mathcal{G})$ where $E$ is an equivalence relation on $Ω$ and $\mathcal{G}$ is a sub-$σ$-algebra of $\mathcal{G}$; we say that $(E,\mathcal{F})$ satisfies "strong duality" if $E$ is $(\mathcal{F}\otimes\mathcal{F})$-measurable and if for all probability measures $\mathbb{P},\mathbb{P}'$ on $(Ω,\mathcal{F})$ we have $$\max_{A\in\mathcal{G}}\vert \mathbb{P}(A)-\mathbb{P}'(A)\vert = \min_{\tilde{\mathbb{P}}\inΠ(\mathbb{P},\mathbb{P}')}(1-\tilde{\mathbb{P}}(E)),$$ where $Π(\mathbb{P},\mathbb{P}')$ denotes the space of couplings of $\mathbb{P}$ and $\mathbb{P}'$, and where "max" and "min" assert that the supremum and infimum are in fact achieved. The results herein give wide sufficient conditions for strong duality to hold, thereby extending a form of Kantorovich duality to a class of cost functions which are irregular from the point of view of topology but regular from the point of view of descriptive set theory. The given conditions recover or strengthen classical results, and they have novel consequences in stochastic calculus, point process theory, and random sequence simulation.
