Bounds on the Distribution of a Sum of Two Random Variables: Revisiting a problem of Kolmogorov with application to Individual Treatment Effects

TL;DR

This paper revisits Kolmogorov's problem of characterizing the distribution of a sum of two random variables with given marginals, clarifies previous bounds, corrects misconceptions, and applies these insights to improve inference on individual treatment effects.

Contribution

The authors provide a corrected and unified characterization of the bounds on the sum's distribution, resolving open issues and clarifying sharpness, especially for discontinuous marginals, with applications to treatment effect inference.

Findings

Corrected bounds on the sum's distribution function.

Clarified the sharpness of bounds for discontinuous marginals.

Improved inference methods for individual treatment effects.

Abstract

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 first given by Markarov (1982) and R\"uschendorf (1982) independently. Frank et al. (1987) provided a solution to the same problem using copula theory. However, though these authors 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 the results due to Fan and Park (2010) concerning individual treatment effects, and to fill some other logical gaps.