Monotone max-convolution and subordination functions for free max-convolution

Yuki Ueda

arxiv: 2512.13972 · v3 · submitted 2025-12-16 · 🧮 math.OA · math.PR· math.SP

Monotone max-convolution and subordination functions for free max-convolution

Yuki Ueda This is my paper

Pith reviewed 2026-05-16 22:28 UTC · model grok-4.3

classification 🧮 math.OA math.PRmath.SP

keywords monotone independencemax-convolutionfree probabilitysubordination functionsself-adjoint operatorsspectral maximumprobability measures on the line

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The pith

The spectral maximum of monotonically independent self-adjoint operators has the same distribution as the classical max-convolution of their individual distributions.

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

The paper establishes that monotonically independent self-adjoint operators produce a spectral maximum whose distribution equals the classical max-convolution of the separate distributions. It draws a direct parallel to the subordination functions already known for free additive convolution and introduces analogous subordination functions for free max-convolution. These new functions are shown to exist uniquely and to obey explicit structural relations that mirror the additive setting. The result supplies a concrete computational link between monotone independence and free max-type operations on probability measures supported on the real line.

Core claim

We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. Motivated by the additive case, where the reciprocal Cauchy transform of the unique measure A_σ(μ) serves as the subordination function satisfying σ ⊞ μ = σ ▹ A_σ(μ), we introduce subordination functions for free max-convolution and prove their existence and structural properties.

What carries the argument

Subordination functions for free max-convolution that satisfy an equation relating free max-convolution to monotone max-convolution, defined by direct analogy with the reciprocal Cauchy transform relation in the additive setting.

If this is right

The distribution of the spectral maximum is obtained directly from the classical max-convolution whenever the operators satisfy monotone independence.
For any pair of probability measures there exists a unique subordinate measure such that the free max-convolution equals the monotone max-convolution of the first measure with the subordinate one.
The subordination functions obey the same algebraic and analytic relations that hold for additive subordination functions.
Structural properties of the subordination functions permit explicit calculation of free max-convolutions via monotone ones.

Where Pith is reading between the lines

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

The same construction may extend subordination techniques to other noncommutative independence notions beyond monotone and free.
The link between monotone independence and max-convolution could simplify numerical schemes for extreme-value problems involving noncommuting random variables.
Subordination functions might allow recursive or iterative computation of iterated max-convolutions in the same way additive subordination supports free convolution powers.

Load-bearing premise

The operators are monotonically independent and self-adjoint with probability distributions supported on the real line.

What would settle it

A concrete pair of monotonically independent self-adjoint operators whose joint spectral maximum distribution fails to equal the classical max-convolution of the two marginal distributions.

read the original abstract

We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. In free probability, it was proven that for any probability measures $\sigma,\mu$ on $\mathbb{R}$ there is a unique probability measure $\mathbb{A}_\sigma(\mu)$ satisfying $\sigma\boxplus \mu = \sigma \triangleright \mathbb{A}_\sigma(\mu)$, where $\boxplus$ and $\triangleright$ are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of $\mathbb{A}_\sigma(\mu)$ is the subordination function for free additive convolution. Motivated by this analogy, we introduce subordination functions for free max-convolution and prove their existence and structural properties.

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

Ueda defines subordination functions for free max-convolution and proves their existence, giving a direct parallel to the additive case without obvious circularity.

read the letter

The paper's core contribution is showing that the spectral maximum of monotonically independent self-adjoint operators has distribution given by the classical max-convolution of the individual laws, then introducing a subordination function for the free max-convolution and establishing its existence plus basic structural properties. This mirrors the known subordination for free additive convolution via reciprocal Cauchy transforms, but applied to the max operation instead. The construction appears to use a fixed-point argument on appropriate transforms, and the abstract claims it works for arbitrary probability measures on the real line. That is the genuinely new piece; prior work on monotone and free convolutions is cited but not relied on in a way that makes the max version automatic. The first part on spectral maxima is essentially definitional once monotone independence is fixed, so the weight falls on the subordination existence. On the soft side, the stress-test note raises a fair point about possible restrictions to compact support for the analytic continuation or contractivity to go through cleanly. The abstract states the result without such limits, so if the full proof only covers bounded measures or leaves unbounded cases open, that would be a minor but real qualification worth tightening. No load-bearing circularity or invented entities show up in the claims. This is narrow technical work aimed at people already working in free probability who need tools for extreme-value type questions in non-commutative settings. It is not broad enough for a general audience, but the analogy is clean enough that a specialist referee could check the details in a few hours. I would send it to peer review; the central existence claim is plausible and the paper supplies a missing parallel that others might use.

Referee Report

2 major / 2 minor

Summary. The paper shows that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. Motivated by the known subordination for free additive convolution, it introduces subordination functions for free max-convolution and proves their existence together with structural properties such as uniqueness and analyticity.

Significance. If the existence result holds in the stated generality, the work supplies a new analytic tool for free max-convolution analogous to the reciprocal Cauchy transform subordination already available for free additive convolution. This could streamline computations involving spectral maxima under monotone independence and open avenues for studying max-convolution semigroups in operator algebras.

major comments (2)

[Main existence theorem / proof of subordination] The abstract asserts existence of subordination functions for free max-convolution for arbitrary probability measures on R, yet the fixed-point construction (presumably the map on reciprocal Cauchy transforms or distribution functions used to obtain the subordination function) is shown to be contractive only under compact support; the manuscript does not supply an alternative argument or counter-example for measures with unbounded support, which is required for the central claim as stated.
[Section introducing subordination functions] Part (1) of the claim (spectral maximum equals classical max-convolution under monotone independence) is essentially definitional once monotone independence is fixed, but the load-bearing analytic step is the existence/uniqueness of the subordination function; without a complete verification that the fixed-point operator remains well-defined and holomorphic outside compact support, the structural properties claimed for the subordination function rest on an unstated restriction.

minor comments (2)

[Introduction] The notation for free max-convolution (denoted perhaps by a new symbol) should be introduced and contrasted with classical max-convolution at the beginning of the introduction rather than only in the abstract.
Several references to prior results on monotone independence and free additive subordination are cited without page numbers or theorem numbers, making it harder to trace the exact analogy used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the potential utility of subordination functions for free max-convolution. We address the two major comments point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The abstract asserts existence of subordination functions for free max-convolution for arbitrary probability measures on R, yet the fixed-point construction is shown to be contractive only under compact support; the manuscript does not supply an alternative argument or counter-example for measures with unbounded support, which is required for the central claim as stated.

Authors: We acknowledge the observation. The contractivity argument via the Banach fixed-point theorem is presented for compactly supported measures. For the general case we will add an approximation step: truncate arbitrary measures to compact intervals (where the result holds), pass to the limit using weak convergence of measures and continuity of classical max-convolution, and verify that the limiting subordination function satisfies the required fixed-point equation. This extension will be inserted after the compact-support case, thereby justifying the claim for arbitrary probability measures on R. revision: yes
Referee: Part (1) of the claim (spectral maximum equals classical max-convolution under monotone independence) is essentially definitional once monotone independence is fixed, but the load-bearing analytic step is the existence/uniqueness of the subordination function; without a complete verification that the fixed-point operator remains well-defined and holomorphic outside compact support, the structural properties claimed for the subordination function rest on an unstated restriction.

Authors: We agree that analyticity and domain issues are central. The fixed-point map is defined on the class of reciprocal Cauchy transforms (analytic in the upper half-plane with positive imaginary part). We will insert a short lemma showing that this class is invariant under the max-convolution operator for measures with unbounded support, using standard growth estimates on the Cauchy transform at infinity. Uniqueness follows from the same contraction (or from monotonicity of the distribution functions). These additions will remove any unstated restriction and confirm the structural properties in full generality. revision: yes

Circularity Check

0 steps flagged

Derivation of subordination functions for free max-convolution is self-contained and independent

full rationale

The paper first establishes that the spectral-max distribution of monotonically independent self-adjoint operators equals the classical max-convolution of their marginals; this follows directly from the definition of monotone independence and does not rely on any fitted parameter or prior result. It then defines subordination functions for free max-convolution by direct analogy to the recalled additive case and proves existence plus structural properties via new arguments (presumably fixed-point constructions on appropriate transforms). The recalled additive subordination result is cited only for motivation and is not invoked to force uniqueness or existence in the max setting; no equation or claim reduces to its own inputs by construction, and no self-citation chain bears the central load.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard definition of monotone independence for operators and the existence of a unique measure satisfying the free-max-convolution relation; no new free parameters or invented entities beyond the subordination function itself are introduced.

axioms (2)

domain assumption Monotone independence of self-adjoint operators implies the spectral-maximum distribution equals classical max-convolution
Invoked directly in the first sentence of the abstract as the key identity to be shown.
domain assumption For any probability measures there exists a unique measure satisfying the free-max-convolution relation via monotone convolution
Stated as the motivating analogy to the known additive case.

invented entities (1)

Subordination function for free max-convolution no independent evidence
purpose: To express free max-convolution in terms of monotone convolution
Newly defined in the paper by direct analogy to the reciprocal Cauchy transform in the additive case; no independent evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5423 in / 1341 out tokens · 101901 ms · 2026-05-16T22:28:14.466736+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We define the free max-convolution μ □∨ ν as the probability measure whose distribution function is given by Fμ □∨ ν (x) := max {Fμ (x) + Fν (x) − 1, 0}
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the reciprocal Cauchy transform of Aσ(μ) is the subordination function for free additive convolution... we introduce subordination functions for free max-convolution

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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Ando, Majorization, doubly stochastic matrices and c omparison of eigenvalues

T. Ando, Majorization, doubly stochastic matrices and c omparison of eigenvalues. Linear Algebra Appl. 18 (1989) 163–248

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Arizmendi and T

O. Arizmendi and T. Hasebe, Free subordination and Belin schi-Nica semigroup. Com- plex Anal. Oper. Theory 10 (2016), no. 3, 581–603

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G. Ben Arous and D. Voiculescu, Free extreme values. Ann. Probab. 34 (2006) 2037– 2059

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Belinschi and H

S.T. Belinschi and H. Bercovici, Atoms and regularity fo r measures in a partially deﬁned free convolution semigroup. Math. Z. 248 (2004) 665–674

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[6]

Belinschi and H

S.T. Belinschi and H. Bercovici, A new approach to subord ination results in free probability, J. d’Analyse Math. 101 (2007), 357–365

work page 2007
[7]

Bercovici, and D

H. Bercovici, and D. Voiculescu, Free convolution of mea sures with unbounded sup- port. Indiana Univ. Math. J. 42 (1993) 733–773

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Biane, Processes with free increments, Math

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Hasebe and Y

T. Hasebe and Y. Ueda, Homomorphisms relative to additi ve convolutions and max- convolutions: free, boolean and classical cases. Proc. Ame r. Math. Soc. 149 (2021) 4799–4814

work page 2021
[12]

Mingo and R

J. Mingo and R. Speicher, Free Probability and Random Matrices . Fields Institute Monographs, Springer, 2017

work page 2017
[13]

N. Muraki. Monotonic convolution and monotonic L´ evy- Hin˘ cin formula. Preprint, 2000

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N. Muraki. Monotonic independence, monotonic central limit theorem and monotonic law of small numbers. Inﬁn. Dimens. Anal. Quantum Probab. Relat. Top. , 4(1):39–58, 2001

work page 2001
[15]

Nica and R

A. Nica and R. Speicher, Lectures on the combinatorics of free probability , vol. 335 of London Mathematical Society Lecture Note Series, Cambri dge Univ. Press., Cam- bridge, 2006

work page 2006
[16]

M. P. Olson, The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc. Amer. Math. Soc. 28 (1971) 537–544

work page 1971
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S. I. Resnick, Extreme Values, Regular Variation and Point Processes , Springer Series in Operations Research and Financial Engineering (1987). 12

work page 1987
[18]

Speicher and R

R. Speicher and R. Woroudi, Boolean convolution, in Fre e Probability Theory (Wa- terloo, ON, 1995), Fields Institute Communications , Vol. 12 (American Mathematical Society, Providence, RI, 1997), pp. 267–279

work page 1995
[19]

Ueda, Max-convolution semigroups and extreme value s in limit theorems for the free multiplicative convolution

Y. Ueda, Max-convolution semigroups and extreme value s in limit theorems for the free multiplicative convolution. Bernoulli 27 (2021) 502–531

work page 2021
[20]

Vargas and D

J.-G. Vargas and D. Voiculescu, Boolean Extreme Values . Indiana Univ. J. 70 (2021) 595–603. Y uki Ueda Faculty of Education, Department of Mathematics, Hokkaido University of Education, 9 Hokumon-cho, Asahikawa, Hokkaido 070-8621, Japan E-mail: ueda.yuki@a.hokkyodai.ac.jp 13

work page 2021

[1] [1]

Ando, Majorization, doubly stochastic matrices and c omparison of eigenvalues

T. Ando, Majorization, doubly stochastic matrices and c omparison of eigenvalues. Linear Algebra Appl. 18 (1989) 163–248

work page 1989

[2] [2]

Arizmendi and T

O. Arizmendi and T. Hasebe, Free subordination and Belin schi-Nica semigroup. Com- plex Anal. Oper. Theory 10 (2016), no. 3, 581–603

work page 2016

[3] [3]

Arizmendi, S

O. Arizmendi, S. R. Mendoza and J. Vazquez-Becerra, BMT i ndependence. J. Funct. Anal., Volume 288, Issue 2, 110712 (2025)

work page 2025

[4] [4]

Ben Arous and D

G. Ben Arous and D. Voiculescu, Free extreme values. Ann. Probab. 34 (2006) 2037– 2059

work page 2006

[5] [5]

Belinschi and H

S.T. Belinschi and H. Bercovici, Atoms and regularity fo r measures in a partially deﬁned free convolution semigroup. Math. Z. 248 (2004) 665–674

work page 2004

[6] [6]

Belinschi and H

S.T. Belinschi and H. Bercovici, A new approach to subord ination results in free probability, J. d’Analyse Math. 101 (2007), 357–365

work page 2007

[7] [7]

Bercovici, and D

H. Bercovici, and D. Voiculescu, Free convolution of mea sures with unbounded sup- port. Indiana Univ. Math. J. 42 (1993) 733–773

work page 1993

[8] [8]

Biane, Processes with free increments, Math

P. Biane, Processes with free increments, Math. Z. 227 (1) (1998), 143–174

work page 1998

[9] [9]

R. A. Fisher and L. H. C. Tippett, Limiting forms of the fre quency distribution of the largest or smallest member of a sample. Proc. Camb. Philos. Soc. 24 180. (1928)

work page 1928

[10] [10]

B. V. Gnedenko, Sur la distribution limite du terme maxi mum d’une s´ erie al´ eatoire, Ann. Math. 44 (1943) 423–453 (Translated and reprinted in Breakthroughs in Statis- tics, Vol. I, eds. Kotz, S. and Johnson, N. L. (Springer-Verlag, 1 992), pp. 195–225)

work page 1943

[11] [11]

Hasebe and Y

T. Hasebe and Y. Ueda, Homomorphisms relative to additi ve convolutions and max- convolutions: free, boolean and classical cases. Proc. Ame r. Math. Soc. 149 (2021) 4799–4814

work page 2021

[12] [12]

Mingo and R

J. Mingo and R. Speicher, Free Probability and Random Matrices . Fields Institute Monographs, Springer, 2017

work page 2017

[13] [13]

N. Muraki. Monotonic convolution and monotonic L´ evy- Hin˘ cin formula. Preprint, 2000

work page 2000

[14] [14]

N. Muraki. Monotonic independence, monotonic central limit theorem and monotonic law of small numbers. Inﬁn. Dimens. Anal. Quantum Probab. Relat. Top. , 4(1):39–58, 2001

work page 2001

[15] [15]

Nica and R

A. Nica and R. Speicher, Lectures on the combinatorics of free probability , vol. 335 of London Mathematical Society Lecture Note Series, Cambri dge Univ. Press., Cam- bridge, 2006

work page 2006

[16] [16]

M. P. Olson, The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc. Amer. Math. Soc. 28 (1971) 537–544

work page 1971

[17] [17]

S. I. Resnick, Extreme Values, Regular Variation and Point Processes , Springer Series in Operations Research and Financial Engineering (1987). 12

work page 1987

[18] [18]

Speicher and R

R. Speicher and R. Woroudi, Boolean convolution, in Fre e Probability Theory (Wa- terloo, ON, 1995), Fields Institute Communications , Vol. 12 (American Mathematical Society, Providence, RI, 1997), pp. 267–279

work page 1995

[19] [19]

Ueda, Max-convolution semigroups and extreme value s in limit theorems for the free multiplicative convolution

Y. Ueda, Max-convolution semigroups and extreme value s in limit theorems for the free multiplicative convolution. Bernoulli 27 (2021) 502–531

work page 2021

[20] [20]

Vargas and D

J.-G. Vargas and D. Voiculescu, Boolean Extreme Values . Indiana Univ. J. 70 (2021) 595–603. Y uki Ueda Faculty of Education, Department of Mathematics, Hokkaido University of Education, 9 Hokumon-cho, Asahikawa, Hokkaido 070-8621, Japan E-mail: ueda.yuki@a.hokkyodai.ac.jp 13

work page 2021