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

TL;DR

We address Kolmogorov's problem of bounding the distribution of the sum $Z=X+Y$ given fixed marginals $F$ and $G$ by deriving sharp, potentially attainable Makarov-type bounds using copula methods and Sklar's theorem. The paper clarifies achievability versus sharpness, corrects prior results for discontinuous marginals, and extends the framework to the difference $X-Y$ with mass-corrected bounds that rectify earlier inaccuracies. It further develops necessary and sufficient conditions for when bounds are achievable, and shows that the upper bound is universally attainable while the lower bound may fail in general. Finally, the results are applied to causal inference for individual treatment effects $\Delta=Y_1-Y_0$, delivering sharper bounds on the distribution of the ITE and illustrating practical implications with numerical examples.

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 Rüschendorf (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.

Figures (4)

• Figure 1: Support of $C_t$ and mass assigned by $C_t$
• Figure 2: Copula $C$ and the image of $x+y=z$ under the $(F,G)$ mapping.
• Figure 3: A zoom in part of Figure \ref{['fig_a_panel']} with rectangle $[m,n]\times[c,d]$ colored in blue.
• Figure 4: In the right two panels of Example \ref{['ex:two']}, the dashed lines show the bounds $\rho_W(F,G)(z)$ and $\tau_W(F,G)(z)$ and the solid lines show what is achieved under the copulas; the difference indicates the bounds are not achievable. Hence only the upper bound on $P(X+Y\leq z)$ and the lower bound on $P(X+Y\leq z)$ can be achieved for all $z$. The upper and lower bounds for $P(X+Y< z)$ and $P(X+Y<z)$ are the same, which follows from part (iv) of Theorem \ref{['thm:fns_exten_low']} and \ref{['thm:fns_exten_up']}. In Example \ref{['ex:one']}, the upper and lower bounds for $P(X+Y\leq z)$ and $P(X+Y<z)$ are different, all bounds on $P(X+Y\leq z)$ and $P(X+Y<z)$ are achieved but under different copula constructions.

Theorems & Definitions (47)

• Definition 1
• Definition 2: embrechts2013note
• Definition 3
• Proposition 4
• Theorem 5: Sklar 1959
• Theorem 6: frank1987best Theorem 2.14
• Definition 7: Achievability at a point
• Definition 8: Pointwise Best-Possible
• Definition 9: Uniformly Sharp
• Theorem 10: frank1987best Theorem 3.2, nelsen2006introduction Theorem 6.1.2