Sums of two dependent uniform random variables: a characterization by joint tail bounds

arXiv:2610.24197 · math.PR, math.CO, math.OC · Submitted 2026-10-16 · Read on arXiv

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Introduction to the show: ident: Mathematics. The day in mathematics: new theorems, new proofs, and what they open.

Ada: We're looking at the paper: "Sums of two dependent uniform random variables".

Emmy: Here's the gist of the paper. Which laws can the sum of two uniform random variables have when nothing is known about how they depend on each other?

Ada: The attainable laws are exactly those that satisfy a family of explicit joint bounds on sums of tail probabilities, one bound for each finite set of thresholds For laws with a density at most one, convex order alone decides The number of thresholds a law needs is set by the spacing of its atoms, not by how many atoms it has.

Emmy: Characterization via Joint Tail Bounds The paper characterizes the set of attainable laws for the sum of two uniform random variables, showing that while convex order is necessary, it is not sufficient, and a family of explicit joint bounds on tail probabilities is required.

Ada: For thresholds 0 < η1 ≤ · · · ≤ ηr < 1, the attainability condition involves a sum over all pairs of thresholds where the sum of their indices equals l + 1, specifically ∑2rl=1 P(S ≥ ξl) ≤ 2∑rq=1 (1 − ηq).

Title and authors: Ada: So, to go over the title and authors again, we have "Sums of two dependent uniform random variables: a characterization by joint tail bounds," written by Prashant Chaudhary and Kapil Khandelwal >

Emmy: I see the title suggests they are looking for a concrete way to characterize exactly which laws on this sum are possible, beyond just the general convex order idea >

Kolya: And what they do is replace that general idea with these joint bounds on tail probabilities, which they show completely describe the set of attainable laws >

Ada: It’s a shift from a global condition to a local one; instead of checking if one distribution is less than another everywhere, you check specific pairs of tail probabilities simultaneously >

Emmy: This feels like taking the structure of the problem and finding an algebraic way to state it using these inequalities, which is what Kantorovich duality helps with >

Kolya: And Kolya on the numbers—they show that for thresholds eta one through eta r, they need a sum over all pairs where the sum of their indices equals l plus one, specifically sum 2r =one P(S at least xi) at most two sum rq=one (one - eta q) > <ref:2610.24197#pg2>

Ada: That specific bound is sharp for every threshold choice they pick, and that’s a pretty strong statement about the attainability condition >

The paper's summary: Emmy: So, looking at the summary of "Sums of two dependent uniform random variables: a characterization by joint tail bounds," the main point is that while convex order isn't sufficient for two variables, there’s a set of explicit joint bounds that *is* sufficient >

Kolya: They are characterizing exactly which laws belong to D U two by these bounds, and they show that for densities bounded below near one, convex order implies the simplest of these joint bounds > <ref:2610.24197#pg2,for densities bounded below near>

Ada: But for densities at most two, convex order only implies the simplest bound, and you need those more complex family of explicit bounds to really characterize the set >

Emmy: The paper is focused on showing that distinct thresholds are enough to decide attainability, which is a big structural simplification compared to needing an infinite number of constraints >

Kolya: And they connect this structure to lattice laws on (one/k)Z, showing that these laws are dense in D U two at a sharp rate in the infinity-Wasserstein metric > <ref:2610.24197#pg2>

The paper's improvements: Ada: Now, when we look at the improvements suggested by this work, they really focus on how to make these bounds more useful for actual applications >

Emmy: They use Kantorovich duality with one potential to reformulate the membership condition into a family of inequalities, which is crucial for deriving that tail-sum characterization >

Kolya: That duality is key because it lets them get that tail-sum characterization of D U two which is what really gives you the explicit conditions > <ref:2610.24197#pg2>

Ada: They also have this grid picture on (one/k)Z where lattice laws are dense in D U two at a sharp rate, and the facet structure of the X-ray polytope P b k dictates the minimal list of tail bounds needed for those lattice laws > <ref:2610.24197#pg2>

Emmy: For mean-one laws on (a, b)Z, they show that this structure is determined by specific potentials and hinges, which gives a very concrete way to look at these complex constraints >

Conclusion: Ada: So to wrap up the paper "Sums of two dependent uniform random variables: a characterization by joint tail bounds," it shows that we need explicit joint bounds on tail probabilities rather than just convex order for two variables >

Emmy: And that these bounds are sharp, and they are necessary because convex order alone isn't enough when the density is two or less >

Kolya: From a probabilistic viewpoint, it tells us that the number of thresholds needed to exclude a law outside D U two is governed by the spacing between atoms, not just how many atoms there are > <ref:2610.24197#pg2>

Ada: It’s important to remember that even though convex order decides things for density one laws, it fails above density two because of those specific joint bounds >

Emmy: And the most interesting open question they leave hanging is whether a density at most two and the constraint S less than or equal to c times two U together imply that the law of S is attainable > <ref:2610.24197#pg2>

Kolya: I agree, that’s the sticking point, and this characterization by joint tail bounds gives us the tools we need to actually test those remaining questions >

University of Notre Dame

math.PR, math.CO, math.OC

Submitted: 2026-10-16

Updated: 2026-10-29

Comments: 25 pages, 1 table; 8-page supplementary material as an ancillary file. Code: https://doi.org/10.5281/zenodo.23231387. Companion paper: arXiv:2610.30058. v2: cites the companion paper by its arXiv identifier and states the two companion results used; code, data and results unchanged

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: Which laws can the sum of two uniform random variables have when nothing is known about how they depend on each other? The attainable laws are exactly those that satisfy a family of explicit joint

Terminology

Sufficiency — 6× in this paper · explained in 2 episodes
Loss Function — 5× in this paper · explained in 2 episodes
Copula — 2× in this paper · explained in 2 episodes
Convex Hull — 1× in this paper · explained in 2 episodes
Monotonicity — 1× in this paper · explained in 2 episodes
Optimal Transport — 1× in this paper · explained in 5 episodes
Quantization — 1× in this paper · explained in 15 episodes

Summary

Which laws can the sum of two uniform random variables have when nothing is known about how they depend on each other? The attainable laws are exactly those that satisfy a family of explicit joint bounds on sums of tail probabilities, one bound for each finite set of thresholds For laws with a density at most one, convex order alone decides The number of thresholds a law needs is set by the spacing of its atoms, not by how many atoms it has.

Characterization via Joint Tail Bounds

The paper characterizes the set of attainable laws for the sum of two uniform random variables, showing that while convex order is necessary, it is not sufficient, and a family of explicit joint bounds on tail probabilities is required. For thresholds 0 < η1 ≤ · · · ≤ ηr < 1, the attainability condition involves a sum over all pairs of thresholds where the sum of their indices equals l + 1, specifically ∑2rl=1 P(S ≥ ξl) ≤ 2∑rq=1 (1 − ηq). This bound is sharp for every η.

The Role of Convex Order and Duality

Convex order alone decides attainability for laws with a density at most one, but at density two, it implies simpler joint bounds. The paper introduces Kantorovich duality with one potential to reformulate the membership condition into a family of inequalities. This duality is crucial for deriving the tail-sum characterization of D U2.

Grid Structures and Lattice Laws

The analysis develops a grid picture on (1/k)Z, where lattice laws are dense in D U2 at a sharp rate in the ∞-Wasserstein metric. The facet structure of the X-ray polytope P b(k) dictates the minimal list of tail bounds needed for lattice laws. For mean-one laws on (a, b)Z, the structure is determined by specific potentials and hinges.

Obstructions to Attainability

The paper investigates what governs the number of thresholds needed to exclude a law outside D U2. For two-point laws, the required number of steps is unbounded, governed by the spacing rather than the number of atoms. The most important open question is whether a density at most 2 and S ≤cx 2U together imply that law(S) is attainable.

Laws with Bounded Density

For laws with a bounded density, convex order decides membership at density one, fails above density two, and for single steps the cutoff is exactly two. The analysis involves constructing specific certificates using potentials that depend on the gaps between thresholds. The result shows that for certain densities and spacing conditions, a law is attainable, while others are not.

Concluding Remarks

The findings indicate that the number of steps needed to exclude a law outside D U2 is not bounded by any function of its number of atoms. The most important open question remains whether a density at most 2 and S ≤cx 2U together imply that law(S) is attainable. The characterization by joint tail bounds provides the necessary tools to address these remaining questions.

Problem 5.1

The problem of whether a density at most 2 and S ≤cx 2U together imply that law(S) is attainable is reduced to extreme densities, as follows. The certificates of Lemma 5.1 are complete, and Problem 5.1 reduces to extreme densities, as follows.

Proposition 5.3

The law of Proposition 5.3 with c = 2 (η = h = 1/4) satisfies P(S ≥ 1/2) + P(S ≥ 5/4) = 3/2 = 2(1 − 1/4).

Theorem 5.3

The law with density c on the 2r intervals [ξl, ξl + δ] has mean one and is dominated by 2U in convex order, and it violates (4.1) by c−r−1/2c(2r+1).

Theorem 5.4

The law of Proposition 5.3 at c = 2 is attainable, so the single-step bound of Proposition 5.2 holds with equality at η = 1/4.

Corollary 4.1

For every r ≥ 1, the laws on [0, 2] with mean one that satisfy (3.1) for every potential with at most r unit steps form a set strictly larger than D U2; it contains the two-point laws Fw,β with r + 1 < 1/β < r + 2.

Corollary 3.2

A mean-one law on (a, a+b)Z belongs to D U2 if and only if k = 1/b ∈ N and, with C = (a + b − 1)/b in (0, 1), f2 ∈ [C(k mod 2)/k, C].

Theorem 4.3

The least number of unit steps, counted with multiplicity, of a potential that excludes a law outside D U2 is max(1, ⌊1/β⌋).

Remark 4.2

The number of thresholds needed to exclude a law outside D U2 is governed by the spacing, not the number of atoms. The grid picture, in which a law on (1/k,..., (2k − 1)/k) is decided by thresholds in (1/k,..., (k − 1)/k), is key.

Example 4.2

The law Fw has masses (1/3, 1/6, 1/6, 1/4, 5/3) on (1/3,..., 5/3); its violation is at least that for distinct thresholds in (a).

Appendix A

The printed bound f2 ≥ a + b − 1/2 should read f2 ≥ a + b - 1. The corrected vertex is exactly (1/18, 1/18, 8/9). The mass f3 sits at a − b, while the statement and the mean identity f3 = f1 + (1 − a)/b put f1 at a − b. The middle mass, and hence the result, is unaffected. The bibliography entry of [1] for Mao and Wang (2015) gives J. Multivariate Anal. 136, 12–25. The paper is [3]. The bibliography entry of [8] for Kellerer, H. G.(1984) gives Z. Wahrsch. Verw. Gebiete 67, 399–432.

Data

The code and data that check every computational statement of this paper are deposited at Zenodo, doi:10.5281/zenodo.23231387. The supplementary material (Sections S1 to S5) is posted with this article as the ancillary file supplement.pdf <ref:2610.

Improvements for AI systems

Improved Risk Aggregation Under Uncertainty: The system can bound risk when marginal laws are known but dependence is unknown by using explicit joint bounds on tail probabilities, replacing convex order with a family of thresholds.
Enhanced Dependence Characterization: The AI can distinguish between attainability and non-attainability for sums of two uniform random variables by checking if the law satisfies the joint tail bounds derived from potential inequalities like (3.1) and (4.1).
Lattice Law Verification: For laws supported on a grid, the system can verify membership in the set D U 2 by checking if a vector with specific entries lies within the polytope of anti-diagonal X-rays P b k, which is equivalent to checking facet inequalities I(u) as described in Proposition 3.1.
Density Bound Verification: For laws with a density at most one, the system can determine attainability solely by checking convex order, while for densities at most two, it verifies that convex order implies all single-step and two-step bounds derived from the family of joint tail bounds.
Unbounded Dependence Detection: The system can detect when dependence is too strong to be characterized by a fixed number of steps by analyzing the spacing of atoms; it can determine that the number of thresholds needed grows as the spacing shrinks, whatever the number of atoms.