We revisit the following problem, proposed by Kolmogorov: given prescribed marginal distributions $F$ and $G$ for random variables $X,Y$ respectively, characterize the set of compatible distribution functions for the sum $Z=X+Y$. Bounds on the distribution function for $Z$ were given by Markarov (1982), and Frank et al. (1987), the latter using copula theory. However, though they obtain the same bounds, they make different assertions concerning their sharpness. In addition, their solutions leave some open problems in the case when the given marginal distribution functions are discontinuous. These issues have led to some confusion and erroneous statements in subsequent literature, which we correct. Kolmogorov's problem is closely related to inferring possible distributions for individual treatment effects $Y_1 - Y_0$ given the marginal distributions of $Y_1$ and $Y_0$; the latter being identified from a randomized experiment. We use our new insights to sharpen and correct results due to Fan and Park (2010) concerning individual treatment effects, and to fill some other logical gaps.

翻译：我们重访以下由柯尔莫哥洛夫提出的问题：给定随机变量$X$和$Y$的边缘分布$F$与$G$，刻画其和$Z=X+Y$的相容分布函数集合。Markarov（1982）与Frank等（1987）分别给出了$Z$的分布函数界限，后者采用了copula理论。然而，尽管他们得到相同的界限，却对其锐利性做出了不同的断言。此外，当给定的边缘分布函数不连续时，他们的解法遗留了若干开放问题。这些争议导致后续文献出现混淆与错误论述，本文对此予以纠正。柯尔莫哥洛夫问题与基于随机实验中可识别的$Y_1$和$Y_0$边缘分布推断个体处理效应$Y_1-Y_0$的可能分布密切相关。我们利用新见解完善并修正了Fan与Park（2010）关于个体处理效应的结论，填补了其他逻辑空白。