arxiv: 2605.10491 · v1 · submitted 2026-05-11 · 🧮 math.PR · math.ST· stat.TH

Recognition: no theorem link

Zero-couplings of infinite measures with cyclically monotone support and multivariate regular variation

Alexandre Reber, Anne Sabourin, Cees de Valk, Johan Segers

Pith reviewed 2026-05-12 05:00 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH

keywords zero-couplingcyclically monotone transportinfinite measuresM_0(R^d)multivariate regular variationexponent measuresBrenier-McCann theorem

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The pith

Cyclically monotone zero-couplings between measures of possibly infinite mass are unique under a Hausdorff dimension condition, extending the Brenier-McCann theorem, and their use in multivariate regular variation yields unique tail limits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a theory of cyclically monotone transport plans for measures in M_0(R^d) that can have infinite total mass and are finite away from the origin. It introduces zero-couplings and proves their existence for any pair of such measures. Under a condition on the Hausdorff dimension of the first measure and when at least one measure has infinite mass, these zero-couplings are shown to be unique. This uniqueness provides an analogue of the classical Brenier-McCann theorem in the infinite-measure setting and allows representation of the couplings via gradients of convex functions. The results are applied to regularly varying probability measures, where cyclically monotone couplings between them have tail limits that match the unique proper zero-coupling between the associated exponent measures.

Core claim

We prove uniqueness of the cyclically monotone zero-coupling, yielding an analogue of the Brenier--McCann theorem in this infinite-measure setting. We further show that a cyclically monotone coupling between two regularly varying distributions admits a tail limit that coincides with the unique proper cyclically monotone zero-coupling between the corresponding exponent measures.

What carries the argument

The cyclically monotone zero-coupling, a transport plan with cyclically monotone support between measures in M_0(R^d) that serves as the analogue of the optimal transport map in the infinite-mass case.

If this is right

Existence holds for arbitrary pairs of measures in M_0 without moment assumptions.
Such couplings admit representation through gradients of closed convex functions.
Under additional conditions the zero-coupling is proper, meaning the second measure is the push-forward of the first via a transport map.
For regularly varying probabilities, the tail limit of any cyclically monotone coupling equals the zero-coupling of the exponent measures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The uniqueness result implies that the transport map is uniquely determined by the measures in the specified class.
The tail limit property provides a way to recover the exponent measure coupling from finite approximations of regularly varying distributions.

Load-bearing premise

The first measure must satisfy a Hausdorff-dimension condition and at least one of the two measures must have infinite mass for uniqueness to hold.

What would settle it

Finding two distinct cyclically monotone zero-couplings for a pair of measures in M_0 where the first measure violates the Hausdorff dimension condition or both measures have finite mass.

read the original abstract

We study cyclically monotone transport plans between measures in $\mathrm{M}_0(\mathbb{R}^d)$, the class of Borel measures on $\mathbb{R}^d \setminus \{0\}$ that are finite on sets bounded away from the origin but may have infinite total mass. We avoid moment assumptions and allow the transport cost to be infinite. This framework naturally arises for exponent measures in multivariate regular variation and includes other examples such as L\'evy measures. We introduce the notion of a zero-coupling and establish existence of cyclically monotone zero-couplings for arbitrary pairs of measures in $\mathrm{M}_0(\mathbb{R}^d)$. Under a Hausdorff-dimension condition on the first measure and when at least one of the two measures has infinite mass, we prove uniqueness of the cyclically monotone zero-coupling, yielding an analogue of the Brenier--McCann theorem in this infinite-measure setting. We further derive a representation of such couplings through gradients of closed convex functions and identify conditions under which the zero-coupling is proper in the sense that the second measure is equal to the restriction to the punctured space of the push-forward of the first measure by a cyclically monotone transport map. Finally, we apply these results to regularly varying probability measures. We show that a cyclically monotone coupling between two such distributions admits a tail limit that coincides with the unique proper cyclically monotone zero-coupling between the corresponding exponent measures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Zero-couplings extend cyclically monotone transport to infinite measures with a uniqueness result under Hausdorff dimension conditions and a tail limit application to regular variation.

read the letter

This paper introduces zero-couplings for measures in M_0(R^d) that can have infinite mass and shows existence of cyclically monotone versions for any pair. It proves uniqueness under a Hausdorff dimension condition on the first measure when at least one has infinite mass, giving a version of the Brenier-McCann theorem here. They represent these couplings as gradients of closed convex functions and specify when they are proper. The key application is that for regularly varying probabilities, cyclically monotone couplings have tail limits matching the unique proper zero-coupling of the exponent measures. The work does a good job extending optimal transport ideas to this infinite setting without moment assumptions, which fits exponent measures and Levy measures. The existence result is broad, and the tail limit consequence follows neatly from the uniqueness. The main limitation is that uniqueness requires the Hausdorff dimension condition, which is stated up front but may not cover every measure one might want to use. Properness is characterized but without many examples it is hard to see how restrictive that is in applications. The abstract outlines the results clearly, and there is no sign of circularity or inconsistency. Since the full proofs are in the manuscript, one would check the convex analysis details for the representation theorem, but the outline matches the stress test with no hidden assumptions. This is for people in probability theory or extreme value statistics who work with infinite measures or regular variation. A reader needing transport maps between such distributions gets new tools. It deserves a serious referee as the claims are specific and build on cited literature in a consistent way. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces zero-couplings for Borel measures in M_0(R^d) that are finite away from the origin but may have infinite total mass. It proves existence of cyclically monotone zero-couplings for arbitrary pairs in this class, establishes uniqueness of the cyclically monotone zero-coupling under a Hausdorff-dimension condition on the first measure together with infinite mass for at least one measure, derives a representation of such couplings via gradients of closed convex functions, identifies conditions for the coupling to be proper, and shows that for regularly varying probability measures a cyclically monotone coupling admits a tail limit that equals the unique proper cyclically monotone zero-coupling of the associated exponent measures. This yields an analogue of the Brenier-McCann theorem in the infinite-measure setting.

Significance. If the stated theorems hold, the work supplies a useful extension of cyclical monotonicity and optimal transport to infinite measures arising in multivariate regular variation and Lévy processes. The uniqueness result and the tail-limit identification for regularly varying laws are the central contributions; they are directly applicable to asymptotic analysis of heavy-tailed distributions and could serve as a foundation for further results on transport maps in this regime.

major comments (2)

[§3] §3 (uniqueness theorem): the proof of uniqueness invokes the Hausdorff-dimension condition on the first measure; a brief discussion or reference showing that the condition is close to necessary (e.g., via a counter-example sketch when the dimension bound fails) would strengthen the claim that the result is a genuine analogue of Brenier-McCann.
[§4] §4 (representation): the statement that every cyclically monotone zero-coupling is the gradient of a closed convex function is central to the proper-coupling identification later used in the regular-variation application; the argument should explicitly verify that the convex function is proper (finite everywhere) under the infinite-mass hypothesis.

minor comments (2)

[Abstract / §2] The definition of a zero-coupling (presumably in §2) is used throughout; a short sentence in the abstract or introduction recalling that it means the coupling has zero cost with respect to the infinite cost function would improve readability for readers outside convex analysis.
[Throughout] Notation for the class M_0(R^d) and for the punctured space is introduced early; ensure that every subsequent statement explicitly recalls whether the measures are restricted to R^d {0} when the push-forward is taken.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address the two major comments below and will incorporate the revisions in the next version of the manuscript.

read point-by-point responses

Referee: [§3] §3 (uniqueness theorem): the proof of uniqueness invokes the Hausdorff-dimension condition on the first measure; a brief discussion or reference showing that the condition is close to necessary (e.g., via a counter-example sketch when the dimension bound fails) would strengthen the claim that the result is a genuine analogue of Brenier-McCann.

Authors: We agree that a short discussion of the necessity of the Hausdorff-dimension condition would strengthen the analogy with the classical Brenier-McCann theorem. In the revised manuscript we will insert a brief remark at the end of Section 3 that recalls where the dimension bound enters the uniqueness argument and sketches a simple counter-example on a lower-dimensional subspace (e.g., when both measures are supported on a line in R^2, multiple distinct cyclically monotone zero-couplings can coexist). revision: yes
Referee: [§4] §4 (representation): the statement that every cyclically monotone zero-coupling is the gradient of a closed convex function is central to the proper-coupling identification later used in the regular-variation application; the argument should explicitly verify that the convex function is proper (finite everywhere) under the infinite-mass hypothesis.

Authors: We thank the referee for pointing out this gap. The infinite-mass assumption is used to guarantee that the convex function remains finite-valued on all of R^d. In the revised Section 4 we will add an explicit paragraph verifying that, when at least one of the two measures has infinite mass, the convex function constructed from the zero-coupling cannot attain the value +∞ at any point, thereby confirming that it is proper in the strong sense required for the subsequent proper-coupling identification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new definitions and proofs

full rationale

The paper introduces the zero-coupling notion and proves existence for arbitrary pairs in M_0(R^d) plus uniqueness under an explicit Hausdorff-dimension condition plus infinite mass, directly yielding the Brenier-McCann analogue. These are established from convex-analysis and measure-theoretic definitions without any reduction to fitted parameters, data-driven predictions, or load-bearing self-citations. The tail-limit application to regularly varying measures follows as a consequence of the uniqueness theorem for exponent measures. No step equates a claimed result to its own inputs by construction; the central claims rest on independent proofs rather than renaming or smuggling prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on standard properties of Borel measures finite away from the origin, cyclically monotone plans from optimal transport theory, and convex analysis; the zero-coupling is the primary new construct.

axioms (2)

domain assumption Borel measures on R^d excluding zero that are finite on sets bounded away from the origin
Definition of the class M_0(R^d) used throughout
standard math Existence and properties of cyclically monotone transport plans
Invoked as background from optimal transport theory

invented entities (1)

zero-coupling no independent evidence
purpose: A cyclically monotone transport plan between infinite measures allowing infinite cost
Newly defined concept whose existence and uniqueness are proved

pith-pipeline@v0.9.0 · 5565 in / 1404 out tokens · 62754 ms · 2026-05-12T05:00:13.579377+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 2 internal anchors

[1]

Albergo, M. S. and E. Vanden-Eijnden (2022). Building normalizing flows with stochastic interpolants.arXiv preprint arXiv:2209.15571

work page internal anchor Pith review arXiv 2022
[2]

Alberti, G. and L. Ambrosio (1999). A geometrical approach to monotone functions in R𝑛.Mathematische Zeitschrift 230, 259–316

work page 1999
[3]

Aleksandrov, A. (1942). Existence and uniqueness of a convex surface with a given integral curvature.C. R. (Doklady) Acad. Sci. URSS(N.S.) 35, 131–134

work page 1942
[4]

Ambrosio, L., E. Brué, D. Semola, et al. (2021).Lectures on optimal transport, Volume 130. Springer

work page 2021
[5]

Anderson, R. D. and V . L. Klee (1952). Convex functions and upper semi-continuous collections.Duke Math. J. 19, 349–357

work page 1952
[6]

Beiglböck, M. (2015). Cyclical monotonicity and the ergodic theorem.Ergodic Theory Dyn. Syst. 35(3)

work page 2015
[7]

Buitendag, E

Beirlant, J., S. Buitendag, E. Del Barrio, M. Hallin, and F. Kamper (2020). Center-outward quantiles and the measurement of multivariate risk.Insurance: Mathematics and Economics 95, 79–100

work page 2020
[8]

Bingham, N. H., C. M. Goldie, J. L. Teugels, and J. Teugels (1989).Regular Variation. Cambridge University Press

work page 1989
[9]

Brenier, Y . (1987). Décomposition polaire et réarrangement monotone des champs de vecteurs.CR Acad. Sci. Paris Sér. I Math. 305, 805–808

work page 1987
[10]

Brenier, Y . (1991). Polar factorization and monotone rearrangement of vector-valued functions.Communications on pure and applied mathematics 44(4), 375–417

work page 1991
[11]

Lavenant, A

Catalano, M., H. Lavenant, A. Lijoi, and I. Prünster (2021). A Wasserstein index of dependence for random measures.arXiv preprint arXiv:2109.06646

work page arXiv 2021
[12]

Galichon, M

Chernozhukov, V ., A. Galichon, M. Hallin, and M. Henry (2017). Monge–kantorovich depth, quantiles, ranks and signs.The Annals of Statistics 45, 223–256

work page 2017
[13]

de Haan, L. and A. Ferreira (2006).Extreme Value Theory: An Introduction. Springer New Y ork

work page 2006
[14]

Kausamo, and K

De Pascale, L., A. Kausamo, and K. Wyczesany (2023). 60 years of cyclic monotonicity: a survey.arXiv preprint arXiv:2308.07682

work page arXiv 2023
[15]

de Valk, C. and J. Segers (2019). Tails of optimal transport plans for regularly varying probability measures. arXiv preprint arXiv:1811.12061v2

work page arXiv 2019
[16]

Figalli, A. and N. Gigli (2010). A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions.Journal de mathématiques pures et appliquées 94(2), 107–130

work page 2010
[17]

Mou, and S

Guillen, N., C. Mou, and S. Swiech (2019). Coupling Lévy measures and comparison principles for viscosity solutions.Transactions of the American Mathematical Society 372(10), 7327–7370. 36

work page 2019
[18]

Hallin, M. (2022). Measure transportation and statistical decision theory.Annual Review of Statistics and Its Application 9(1), 401–424

work page 2022
[19]

Del Barrio, J

Hallin, M., E. Del Barrio, J. Cuesta-Albertos, and C. Matrán (2021). Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach.Annals of Statistics 49, 1139–1165

work page 2021
[20]

Huesmann, M. (2016). Optimal transport between random measures.Ann. Inst. Henri Poincaré Probab. Stat. 52(1), 196–232

work page 2016
[21]

and K.-T

Huesmann, M. and K.-T. Sturm (2013). Optimal transport from Lebesgue to Poisson.The Annals of Probability 41(4), 2426–2478

work page 2013
[22]

Hult, H. and F. Lindskog (2006). Regular Variation for Measures on Metric Spaces.Publ. Inst. Math. 80(94), 121–140

work page 2006
[23]

Lindskog, F., S. I. Resnick, and J. Roy (2014). Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps.Probability Surveys 11, 270–314

work page 2014
[24]

Lipman, Y ., R. T. Chen, H. Ben-Hamu, M. Nickel, and M. Le (2022). Flow matching for generative modeling. arXiv preprint arXiv:2210.02747

work page internal anchor Pith review Pith/arXiv arXiv 2022
[25]

McCann, R. J. (1995). Existence and uniqueness of monotone measure-preserving maps.Duke Mathematical Journal 80(2), 309–323

work page 1995
[26]

(2017).Theory of Random Sets

Molchanov, I. (2017).Theory of Random Sets. Springer Cham

work page 2017
[27]

Pratelli, A. (2008). On the sufficiency of c-cyclical monotonicity for optimality of transport plans.Mathema- tische Zeitschrift 258(3), 677–690

work page 2008
[28]

Resnick, S. I. (2007).Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Science & Business Media

work page 2007
[29]

Resnick, S. I. (2008).Extreme Values, Regular Variation and Point Processes. Springer Science & Business Media

work page 2008
[30]

Resnick, S. I. (2024).The art of finding hidden risks: Hidden regular variation in the 21st century. Springer Nature

work page 2024
[31]

Rockafellar, R. T. (1970).Convex Analysis. Princeton: Princeton University Press

work page 1970
[32]

Rockafellar, R. T. and R. J.-B. Wets (1998).Variational Analysis. New Y ork : Springer

work page 1998
[33]

Rüschendorf, L. and S. T. Rachev (1990). A characterization of random variables with minimum 𝐿2-distance. Journal of Multivariate Analysis 32(1), 48–54

work page 1990
[34]

(2015).Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling

Santambrogio, F. (2015).Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling. Cham: Birkhäuser/Springer

work page 2015
[35]

Schachermayer, W . and J. Teichmann (2009). Characterization of optimal transport plans for the Monge– Kantorovich problem.Proceedings of the American Mathematical Society 137(2), 519–529

work page 2009
[36]

Segers, J. (2022). Graphical and uniform consistency of estimated optimal transport plans.arXiv preprint arXiv:2208.02508

work page arXiv 2022
[37]

(2009).Optimal Transport: Old and New, Volume 338

Villani, C. (2009).Optimal Transport: Old and New, Volume 338. Springer. Appendix A: Background A.1. Portmanteau and Prohorov theorem for infinite measures The spaceM 0(R𝑑) enjoys useful properties similar to the classical Portmanteau theorem or Prohorov theorem for probability measures which will be of constant use. We refer to [ 22,23] for details and q...

work page 2009
[38]

It follows that 𝑆(𝑥)=𝑇(𝑥) for all 𝑥∈ ˆ𝑉

can then be reused directly since it exclusively relies on properties of maximal monotone maps and not on the finiteness of the measures considered there. It follows that 𝑆(𝑥)=𝑇(𝑥) for all 𝑥∈ ˆ𝑉. Since spt𝜇⊂spt ˆ𝜇, we have𝑆(𝑥)=𝑇(𝑥)for all𝑥∈𝑉too. The statements about 𝑊 follow by switching the roles of𝜇 and 𝜈, upon noting thatdom𝑆 −1 =rge𝑆 . Lemma B.7.Let 𝜇...

work page