  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.
