arxiv: 2605.07920 · v1 · submitted 2026-05-08 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

Primitive Sequences for Probability Measures on Compact Intervals

Robert Zimmerman

Pith reviewed 2026-05-11 03:25 UTC · model grok-4.3

classification 🧮 math.PR

keywords primitive sequencesprobability measurescompact intervalscumulative distribution functionsmoment sequenceshomeomorphismsantiderivatives

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The pith

The mapping from probability measures on compact intervals to their primitive sequences is a homeomorphism.

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

The paper defines a primitive sequence for any random variable supported on a compact interval [a, b] by taking the cumulative distribution function, repeatedly antidifferentiating it, and evaluating each antiderivative at the upper endpoint b. This sequence is shown to be identical to a factorially rescaled moment sequence of the reflected variable b minus X, which makes qualitative features of the distribution visible directly in the sequence terms. The central result establishes that the correspondence between probability measures and these sequences is a homeomorphism under appropriate topologies on the two spaces. The construction also yields a characterization of exactly which sequences arise from some probability measure and produces sharp upper and lower bounds on two functionals over all measures that share the same first m sequence entries.

Core claim

We construct primitive sequences by repeated antidifferentiation of the CDF of X evaluated at the upper endpoint b; these sequences are identified with factorially rescaled moment sequences of b-X; the map between measures and sequences is a homeomorphism; admissible sequences are characterized; and sharp bounds are obtained on functionals for measures with fixed initial terms.

What carries the argument

The primitive sequence, obtained by iterated antidifferentiation of the CDF and evaluation at the fixed upper endpoint b, which functions as a complete and topologically faithful representation equivalent to a rescaled moment sequence of the reflected variable.

If this is right

Every probability measure on [a, b] corresponds to a unique primitive sequence and vice versa under the homeomorphism.
A sequence is admissible precisely when it satisfies the conditions derived from classical moment theory applied to the reflected variable.
All measures sharing the first m primitive terms lie in a set whose two chosen functionals attain sharp bounds that can be computed explicitly from those terms.
Qualitative distribution properties such as support width, skewness, or modality are encoded transparently in the decay or sign pattern of the sequence terms.

Where Pith is reading between the lines

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

Truncation of primitive sequences could supply a new method for constructing approximating distributions that respect the homeomorphism.
The link to moment sequences may allow transfer of existing moment inequalities or quadrature rules directly into statements about primitive-sequence-constrained measures.
Because the construction uses only antiderivatives and endpoint evaluation, it may adapt naturally to numerical or functional-analytic settings where CDFs are available only in discretized form.

Load-bearing premise

The random variable must be supported on a compact interval so that the iterated antiderivatives remain well-defined and evaluation at the fixed endpoint produces a complete sequence representation.

What would settle it

Two distinct probability measures on the same compact interval that produce identical primitive sequences would disprove the injectivity (and thus the homeomorphism) of the map.

Figures

Figures reproduced from arXiv: 2605.07920 by Robert Zimmerman.

**Figure 1.** Figure 1: Sharp upper and lower bounds for Fm(1/2), where Fm is supported on [0, 1] and matches the first m moments of the Unif(0, 1) distribution, for m = 1, . . . , 100. 6 Discussion In this paper, we develop an initial theory of the primitive sequence associated with a probability measure supported on a compact interval. In particular, the connection between primitive sequences and Hausdorff moment sequences allo… view at source ↗

read the original abstract

We introduce a sequence representation of a random variable $X$ supported on a compact interval $[a,b]$, which we call a primitive sequence. We construct this sequence by repeatedly antidifferentiating the associated cumulative distribution function of $X$ and evaluating the antiderivatives at the endpoint $b$. We show that the primitive sequence of $X$ can be identified as a factorially rescaled moment sequence of the reflected random variable $b-X$. Through this identification, we show that the primitive sequence transparently captures qualitative features of the distribution of $X$. We then connect primitive sequences directly to classical moment theory and exploit this connection to characterize admissible primitive sequences and to show that under natural topologies, the map from probability measures to primitive sequences is a homeomorphism. We end by examining the set of probability measures whose first $m$ primitive sequence terms are fixed, and thereby obtaining sharp upper and lower bounds on two functionals of those 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

The paper defines primitive sequences via iterated antiderivatives of the CDF on [a,b] and identifies them with rescaled moments of b-X, which imports standard moment results for a homeomorphism and sharp bounds.

read the letter

The main takeaway is that this work gives a sequence representation for probability measures on compact intervals by repeatedly integrating the CDF and evaluating at the right endpoint. That sequence matches the factorial-rescaled moments of the reflected variable b-X, which immediately connects the construction to the Hausdorff moment problem. From there the paper gets a homeomorphism between the space of measures under the weak topology and admissible sequences under the product topology, plus a characterization of which sequences arise and sharp bounds on two functionals when the first m terms are fixed. Those bounds come from the usual extremal discrete measures with at most m+1 points. The link is made cleanly and the qualitative features of the distribution show up directly in the sequence terms, which is the useful part of the representation. The compact support assumption is stated up front and is required for the antiderivatives to be well-defined and for the endpoint evaluation to work; without it the construction would not close. The identification itself follows from standard Fubini applications on a compact interval, so the technical steps are classical rather than novel. The novelty sits in the specific sequence and the way it organizes the qualitative and bounding questions. No circularity or self-referential fitting appears. This is for people working in theoretical probability on compact spaces or in moment problems who want an alternative handle for bounds and qualitative analysis. A reader already comfortable with Hausdorff moments will see the value quickly. The paper deserves a serious referee because the claims rest on verifiable classical facts and the new object is explicitly constructed and motivated.

Referee Report

1 major / 3 minor

Summary. The manuscript introduces primitive sequences for a random variable X supported on a compact interval [a,b], constructed by repeated antidifferentiation of the CDF of X followed by evaluation at the endpoint b. It identifies the resulting sequence as the factorially rescaled moment sequence of the reflected variable b-X via iterated integral representations. The paper then uses this link to classical Hausdorff moment theory to characterize admissible primitive sequences, prove that the map from probability measures (under the weak topology) to primitive sequences (under the product topology) is a homeomorphism, and obtain sharp upper and lower bounds on two functionals over the set of measures with the first m primitive terms fixed.

Significance. If the central identifications and applications hold, the work supplies a transparent integral-based sequence representation that directly encodes distribution features on bounded intervals and inherits the full power of the classical Hausdorff moment problem without introducing new parameters or ad-hoc constructions. The homeomorphism and extremal bounds follow from standard facts (moment map continuity and discrete extremal measures), but the explicit antiderivative construction may prove useful for qualitative analysis or numerical work. Credit is due for the clean Fubini-based identification and for routing all claims through established theory rather than novel axioms.

major comments (1)

The identification of the primitive sequence with rescaled moments (via repeated integration against the CDF) is load-bearing for the homeomorphism and characterization claims; the manuscript should include an explicit inductive formula relating the k-th primitive term to (1/k!) E[(b-X)^k] together with a self-contained verification that the iterated antiderivatives remain well-defined on the compact interval.

minor comments (3)

The abstract refers to 'two functionals' without naming them; the introduction or the section on fixed initial terms should state explicitly what these functionals are (e.g., mean, variance, or other moments) so that the sharpness claim is immediately intelligible.
Notation for the primitive sequence should be introduced once and used consistently; a brief table or displayed equation listing the first three or four terms for a concrete example (uniform or Beta) would help readers verify the construction.
The topology section should cite a standard reference for the fact that the moment map is a homeomorphism onto the admissible Hausdorff sequences (e.g., a textbook treatment of the moment problem) rather than treating it as immediate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestion. We address the major comment below.

read point-by-point responses

Referee: The identification of the primitive sequence with rescaled moments (via repeated integration against the CDF) is load-bearing for the homeomorphism and characterization claims; the manuscript should include an explicit inductive formula relating the k-th primitive term to (1/k!) E[(b-X)^k] together with a self-contained verification that the iterated antiderivatives remain well-defined on the compact interval.

Authors: We agree that an explicit inductive formula and a self-contained verification would strengthen the presentation. While the manuscript already derives the identification via iterated integral representations and Fubini, we will add a dedicated lemma in the revised version that states the inductive relation p_k = (1/k!) E[(b-X)^k] and proves it by induction on k, starting from the base case k=0 (where p_0 = 1) and using the integral representation of each antiderivative. The proof will explicitly invoke Fubini's theorem to interchange the order of integration. We will also include a short paragraph verifying that each iterated antiderivative is well-defined and continuous on the compact interval [a,b], since the CDF is bounded and measurable and all integrals are taken over a finite-length interval. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs the primitive sequence explicitly by iterated antidifferentiation of the CDF followed by evaluation at the fixed endpoint b. The claimed identification with factorially rescaled moments of b-X is obtained by direct repeated application of the integral representation of moments (via Fubini or integration by parts), which is a standard calculation on compact intervals and does not presuppose the final result. The homeomorphism is then deduced from the classical theorem that the Hausdorff moment map is a homeomorphism from Prob([a,b]) (weak topology) to admissible moment sequences (product topology). Admissibility is characterized by the classical positive-semidefiniteness conditions on the shifted Hankel matrices, and the extremal bounds for fixed initial terms are the standard values attained by discrete measures with at most m+1 atoms. No step reduces by definition or by self-citation to its own inputs; all load-bearing facts are external classical results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the newly introduced primitive sequence definition and its equivalence to moments, together with standard properties of CDFs and antiderivatives on compact intervals; no free parameters are introduced and the only invented entity is the sequence itself.

axioms (2)

standard math Cumulative distribution functions are non-decreasing, right-continuous, with F(a)=0 and F(b)=1.
Invoked implicitly to ensure the antiderivatives are well-defined and the sequence encodes the full distribution.
standard math Iterated antiderivatives of integrable functions on compact intervals exist and can be evaluated at the endpoint.
Fundamental theorem of calculus applied repeatedly to the CDF.

invented entities (1)

primitive sequence no independent evidence
purpose: Sequence representation of probability measures on [a,b] obtained by iterated antiderivatives of the CDF evaluated at b.
Newly defined construction whose properties are derived in the paper; no independent external evidence is provided.

pith-pipeline@v0.9.0 · 5444 in / 1596 out tokens · 48884 ms · 2026-05-11T03:25:58.061346+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
We show that the primitive sequence of X can be identified as a factorially rescaled moment sequence of the reflected random variable b−X. ... admissible primitive sequences ... Hausdorff criterion

Reference graph

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