Abstract In this paper we study the maximal possible difference N of values of a quasi-copula at two different points of the unit square. This study enables us to give upper and lower bounds, called constrained bounds, for quasi-copulas with fixed value at a given point in the unit square, thus extending an earlier result from copulas to quasi-copulas. It turns out that the two bounds are actually copulas. Difference N is also the main tool in exhibiting two new characterizations of quasi-copulas, a major result of this paper, which sheds new light on the subject of copulas as well. Significant applications of our results are also given in the imprecise probability theory, one of the more important non-standard approaches to probability. After a full-scale bivariate Sklar's theorem has been proven under this approach, we want to establish the tightness of its background before moving to the more general multivariate scene. We present an extension of the partial order on quasi-distributions used in the said theorem, i.e., pointwise order with fixed margins, using again the difference N as a main tool. A careful study of the interplay between the order on quasi-distributions and the order on corresponding quasi-copulas that represent them is also given. Due to a recent result that the quasi-copulas obtained via Sklar's theorem in the imprecise setting are exactly the same as the ones in the standard setting, it is not surprising that results on quasi-copulas can shed some light both on open questions in the standard probability theory and in the imprecise probability theory at the same time.

中文翻译：

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关于双变量分布的新偏序及其联结的约束边界

摘要 在本文中，我们研究了在单位正方形的两个不同点上的拟联结值的最大可能差值N。这项研究使我们能够为单位正方形中给定点处具有固定值的准 copula 给出上限和下限，称为约束边界，从而将较早的结果从 copula 扩展到准 copula。事实证明，这两个边界实际上是 copula。差异 N 也是展示准 copula 的两个新特征的主要工具，这是本​​文的一个主要结果，它也为 copula 的主题提供了新的思路。我们的结果的重要应用也在不精确概率论中给出，这是概率的更重要的非标准方法之一。在这种方法下证明了全尺度双变量 Sklar 定理之后，我们想在移动到更一般的多元场景之前确定其背景的紧密度。我们提出了上述定理中使用的准分布的偏序的扩展，即具有固定边距的逐点序，再次使用差 N 作为主要工具。还详细研究了准分布上的阶与表示它们的相应准联结上的阶之间的相互作用。由于最近的一个结果是，在不精确设置中通过 Sklar 定理获得的准 copula 与标准设置中的完全相同，因此，准 copula 的结果可以为两个开放问题提供一些启示也就不足为奇了。标准概率论和不精确概率论同时存在。