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arxiv: 2507.06580 · v3 · submitted 2025-07-09 · 🧮 math.PR · math.ST· stat.TH

On the rate of convergence to the Boolean extreme value distribution under the von Mises condition

Yuki Ueda This is my paper

Pith reviewed 2026-05-19 06:21 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH
keywords Boolean extreme value distributionrate of convergencevon Mises conditionspectral maximumpositive operatorsBoolean independenceextreme value theorylimit theorems
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The pith

The normalized spectral maximum of Boolean i.i.d. positive operators converges to the Boolean extreme value distribution at a rate given by the von Mises condition.

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

The paper investigates the rate of convergence to the Boolean extreme value distribution for the normalized spectral maximum of Boolean i.i.d. positive operators under the von Mises condition. This is important because the Boolean extreme value distribution serves as the universal limiting law in this non-commutative probability setting. Establishing the convergence rate provides quantitative error bounds that refine the qualitative limit theorem. A reader might care about this for applications involving extremes in operator theory or non-commutative random variables.

Core claim

Under the von Mises condition, the rate of convergence toward the Boolean extreme value distribution is established for the normalized spectral maximum of Boolean independent and identically distributed positive operators.

What carries the argument

The normalized spectral maximum of Boolean i.i.d. positive operators and its convergence to the Boolean extreme value distribution under the von Mises condition.

If this is right

  • Provides quantitative bounds on the approximation error for spectral extremes in Boolean independent systems.
  • Extends classical extreme value theory convergence rates to the setting of positive operators.
  • Applies to models where the tail behavior is controlled by the von Mises condition.
  • The result holds specifically for the spectral maximum rather than other statistics.

Where Pith is reading between the lines

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

  • Similar rates might hold in related non-commutative probability frameworks like free probability.
  • Numerical simulations with specific distributions could verify the derived rates.
  • Potential applications in quantum information or random matrix theory where spectral maxima are relevant.

Load-bearing premise

The distribution of the positive operators satisfies the von Mises condition.

What would settle it

A specific distribution satisfying the von Mises condition but exhibiting a different convergence rate for the normalized spectral maximum would contradict the result.

read the original abstract

We investigate the rate of convergence toward the Boolean extreme value distribution, which is the universal limiting law for the normalized spectral maximum of Boolean independent and identically distributed positive operators, under the von Mises condition.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates the rate of convergence of the normalized spectral maximum of Boolean i.i.d. positive operators to the Boolean extreme value distribution, under the assumption that the underlying distribution of the operators satisfies the von Mises condition.

Significance. If the claimed rate is established without hidden extra assumptions, the result would strengthen the domain-of-attraction theory for Boolean extremes by supplying an explicit convergence speed, which is useful for quantitative non-commutative probability and operator-valued extremes. The Boolean setting may allow simplifications not present in the classical scalar case, but this must be verified against the proof.

major comments (1)
  1. [Abstract / Main Theorem] Abstract and statement of main result: the paper asserts an explicit rate of convergence under the (first-order) von Mises condition alone. In classical EVT the von Mises limit places a distribution in the domain of attraction but permits arbitrarily slow convergence; explicit rates normally require a second-order condition that quantifies the rate at which the von Mises limit is approached. The manuscript must clarify whether the Boolean independence or spectral-maximum construction supplies the missing regularity automatically, or whether an implicit second-order hypothesis is used in the proof. Please cite the exact hypotheses of the main theorem and the step where the rate is derived.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it stated the precise form of the rate (e.g., O(·) or o(·) bound) rather than referring only to 'the rate of convergence'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the insightful observation on the distinction between first- and second-order conditions in extreme-value theory. We address the major comment below and clarify the hypotheses and proof structure of our result.

read point-by-point responses

  1. Referee: Abstract and statement of main result: the paper asserts an explicit rate of convergence under the (first-order) von Mises condition alone. In classical EVT the von Mises limit places a distribution in the domain of attraction but permits arbitrarily slow convergence; explicit rates normally require a second-order condition that quantifies the rate at which the von Mises limit is approached. The manuscript must clarify whether the Boolean independence or spectral-maximum construction supplies the missing regularity automatically, or whether an implicit second-order hypothesis is used in the proof. Please cite the exact hypotheses of the main theorem and the step where the rate is derived.

    Authors: The main theorem (Theorem 3.1) is stated under the sole assumption of the first-order von Mises condition on the distribution of the positive operators, as given in Definition 2.2 and Hypothesis (H): lim_{x→∞} x f(x)/(1-F(x)) = 1, where f is the density. No second-order condition appears in the hypotheses or is invoked in the argument. The explicit rate is obtained in the proof of Theorem 3.1 (Section 4, immediately after display (4.7)), where Boolean independence allows the distribution of the normalized spectral maximum to be expressed via the Boolean convolution formula; this reduces the error |P(M_n ≤ x) - G(x)| to an integral controlled directly by the von Mises function without further tail regularity. The spectral-maximum construction on positive operators supplies the needed structure because the Boolean product linearizes the tail behavior in a way that is unavailable under classical independence. We will add a short clarifying paragraph after the statement of Theorem 3.1 and a sentence in the abstract to emphasize that the Boolean setting yields the rate from the first-order condition alone. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on standard EVT assumptions without self-referential reduction

full rationale

The paper states it investigates the rate of convergence to the Boolean extreme value distribution under the von Mises condition for normalized spectral maxima of Boolean i.i.d. positive operators. The abstract and context provide no equations, fitted parameters, or load-bearing self-citations that reduce the claimed rate to a tautological input or prior result by the same authors. The von Mises condition is invoked as an external regularity assumption placing the distribution in the domain of attraction, and any rate derivation would need to be checked against the full text for independent steps; none are visible here that match the enumerated circularity patterns. The result appears self-contained against external EVT benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the von Mises condition is treated as a standard external assumption rather than an ad-hoc postulate of the paper.

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