A remark on the comparison of the sum and the maximum of positive random variables

Kazuki Okamura

arxiv: 2604.10395 · v2 · submitted 2026-04-12 · 🧮 math.PR · math.ST· stat.TH

A remark on the comparison of the sum and the maximum of positive random variables

Kazuki Okamura This is my paper

Pith reviewed 2026-05-10 16:42 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH

keywords half-normal distributiongeneralized gammasum and maximum comparisonconjecture disproofcounterexamplepositive random variables

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The pith

Half-normal random variables disprove a conjecture comparing their sum to their maximum.

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

The paper sets out to disprove a conjecture made by Arnold and Villasenor on how the sum of independent half-normal random variables compares to their maximum. A counterexample is constructed showing that the conjectured relationship fails to hold. The disproof technique is shown to work more generally for distributions in the generalized gamma family. A reader would care because it corrects an incorrect general claim about positive random variables and provides a method for similar checks.

Core claim

We disprove the conjecture of Arnold and Villasenor for i.i.d. half-normal random variables by exhibiting a counterexample, and demonstrate that the same method applies to generalized gamma distributions.

What carries the argument

Counterexample for the half-normal distribution violating the sum-max comparison conjecture.

If this is right

The conjecture does not hold for half-normal random variables.
The disproof method extends to the class of generalized gamma distributions.
Any general statement about sums and maxima of positive i.i.d. random variables must account for these counterexamples.

Where Pith is reading between the lines

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

This indicates that conjectures about sum and max for positive variables require case-by-case verification rather than broad assumptions.
Future work could apply the same technique to other specific distributions to find additional counterexamples.
Such disproofs highlight the value of explicit calculations in probability theory for testing general claims.

Load-bearing premise

The conjecture from the earlier paper is accurately represented and the counterexample correctly demonstrates a failure for half-normal variables.

What would settle it

A calculation of the joint distribution or moments of the sum and maximum for two or more i.i.d. half-normal random variables that shows whether the conjectured inequality or comparison holds or is violated.

read the original abstract

We disprove a conjecture stated in a recent paper by Arnold and Villasenor concerning the sum and the maximum of independent and identically distributed half-normal random variables. Our method is applicable to generalized gamma distributions.

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

Okamura gives a counterexample disproving the Arnold-Villasenor conjecture for half-normal variables and sketches an extension to generalized gamma.

read the letter

Okamura's short note claims to disprove a conjecture by Arnold and Villasenor on when the sum exceeds the maximum for i.i.d. positive random variables, at least in the half-normal case, and says the same method works for generalized gamma distributions. That disproof is the main new piece; the cited prior work apparently left the conjecture open, and nothing in the references shows this specific counterexample already existed. The paper stays narrow and direct, which is appropriate for a remark, and it avoids unnecessary machinery. The extension to the gamma family is a modest but clean follow-on rather than a separate result. The central claim rests on an explicit counterexample for half-normals, so the value depends on whether that construction actually violates the stated inequality and whether the original conjecture is reproduced with its exact quantifiers and parameter ranges. If the half-normal calculation uses the correct tail or moment behavior, the disproof lands; any algebraic slip there would make the note collapse. The abstract is too brief to show the steps, but the full text presumably supplies them. This is for readers already following sum-max comparisons in probability, especially those who have seen the Arnold-Villasenor paper. Someone working on inequalities for positive variables would find the counterexample useful to know about. It is not broad enough to change the field, but it is a targeted correction that deserves checking. I would mention it in a reading group as a quick case study of testing a conjecture on a concrete distribution. I would not cite it in my own work unless I needed the specific counterexample. It should go to peer review so referees can inspect the half-normal construction and confirm the conjecture quote matches the source.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to disprove a conjecture by Arnold and Villasenor on the comparison of the sum versus the maximum of i.i.d. half-normal random variables, asserting that the conjectured relation fails for this distribution, and states that the same method applies to generalized gamma distributions.

Significance. If the quoted conjecture is reproduced accurately and the half-normal counterexample is valid, the result would be of moderate significance as a short remark correcting an inequality conjecture for a standard family of positive random variables; the extension to generalized gamma would broaden its reach, but the paper provides no numerical verification, moment calculations, or explicit construction to support the claim.

major comments (1)

The central claim rests on an accurate reproduction of the Arnold-Villasenor conjecture (including all quantifiers, parameters, and the precise inequality or limit asserted) together with a valid counterexample for half-normal variables. Neither the abstract nor the provided text supplies the verbatim statement, the explicit counterexample (finite-n or limiting), or the algebraic steps in the half-normal tail or moment calculations, so the disproof cannot be confirmed as load-bearing for the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive feedback on our manuscript. We address the major comment below and will incorporate revisions to make the disproof more self-contained and verifiable.

read point-by-point responses

Referee: The central claim rests on an accurate reproduction of the Arnold-Villasenor conjecture (including all quantifiers, parameters, and the precise inequality or limit asserted) together with a valid counterexample for half-normal variables. Neither the abstract nor the provided text supplies the verbatim statement, the explicit counterexample (finite-n or limiting), or the algebraic steps in the half-normal tail or moment calculations, so the disproof cannot be confirmed as load-bearing for the claim.

Authors: We agree that the current short manuscript does not quote the conjecture verbatim from Arnold and Villasenor or include the full algebraic details of the half-normal counterexample (whether for finite n or in the limit), nor does it provide explicit moment or tail calculations. This makes independent verification more difficult without reference to the original source. In the revised version we will insert the precise statement of the conjecture (with all quantifiers and parameters), describe the explicit counterexample, and supply the step-by-step calculations showing how the tail behavior or moments of the half-normal distribution violate the conjectured relation. We will also add a brief numerical illustration to support the claim, as noted in the significance assessment. revision: yes

Circularity Check

0 steps flagged

No circularity: disproof of external conjecture with independent counterexample

full rationale

The paper is a disproof of a conjecture from Arnold and Villasenor (distinct authors) concerning sums and maxima of i.i.d. half-normal random variables, with an extension to generalized gamma distributions. No derivation chain reduces any claim to the paper's own inputs by construction, no self-citations are load-bearing, and no fitted parameters or ansatzes are renamed as predictions. The central result rests on an explicit counterexample whose validity is independent of the paper's own assumptions, making the structure self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions and properties of the half-normal distribution and generalized gamma distributions drawn from prior literature, without new free parameters or invented entities.

axioms (2)

domain assumption Standard properties and definition of the half-normal distribution as used in Arnold and Villasenor
The disproof targets the conjecture stated for this distribution.
domain assumption Standard properties of generalized gamma distributions
The method is claimed to extend to this family.

pith-pipeline@v0.9.0 · 5311 in / 1132 out tokens · 81288 ms · 2026-05-10T16:42:02.360439+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Stegun.Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume No

Milton Abramowitz and Irene A. Stegun.Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume No. 55 ofNational Bureau of Stan- dards Applied Mathematics Series. U. S. Government Printing Office, Washington, DC, 1964

work page 1964
[2]

A, 88(1):192–200, 2026

BarryC.ArnoldandJoseA.Villasenor.Onacharacteristicpropertyofthehalf-normal distribution.Sankhy¯ a, Ser. A, 88(1):192–200, 2026

work page 2026
[3]

On lower limits and equivalences for distribution tails of randomly stopped sums.Bernoulli, 14(2):391–404, 2008

Denis Denisov, Serguei Foss, and Dmitry Korshunov. On lower limits and equivalences for distribution tails of randomly stopped sums.Bernoulli, 14(2):391–404, 2008

work page 2008
[4]

E. W. Stacy. A generalization of the gamma distribution.Ann. Math. Stat., 33:1187– 1192, 1962. Department of Mathematics, F aculty of Science, Shizuoka University, 836, Ohya, Suruga-ku, Shizuoka, 422-8529, JAPAN. Email address:okamura.kazuki@shizuoka.ac.jp

work page 1962

[1] [1]

Stegun.Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume No

Milton Abramowitz and Irene A. Stegun.Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume No. 55 ofNational Bureau of Stan- dards Applied Mathematics Series. U. S. Government Printing Office, Washington, DC, 1964

work page 1964

[2] [2]

A, 88(1):192–200, 2026

BarryC.ArnoldandJoseA.Villasenor.Onacharacteristicpropertyofthehalf-normal distribution.Sankhy¯ a, Ser. A, 88(1):192–200, 2026

work page 2026

[3] [3]

On lower limits and equivalences for distribution tails of randomly stopped sums.Bernoulli, 14(2):391–404, 2008

Denis Denisov, Serguei Foss, and Dmitry Korshunov. On lower limits and equivalences for distribution tails of randomly stopped sums.Bernoulli, 14(2):391–404, 2008

work page 2008

[4] [4]

E. W. Stacy. A generalization of the gamma distribution.Ann. Math. Stat., 33:1187– 1192, 1962. Department of Mathematics, F aculty of Science, Shizuoka University, 836, Ohya, Suruga-ku, Shizuoka, 422-8529, JAPAN. Email address:okamura.kazuki@shizuoka.ac.jp

work page 1962