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arxiv: 1907.04672 · v1 · pith:XD4EXWTYnew · submitted 2019-07-10 · 🧮 math.PR

On the q-moment determinacy of probability distributions

Sofiya Ostrovska , Mehmet Turan This is my paper

Pith reviewed 2026-05-24 23:43 UTC · model grok-4.3

classification 🧮 math.PR
keywords q-moment determinacyq-densityprobability distributionsmoment determinacyabsolutely continuous distributionsq-moments
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The pith

New conditions establish q-moment determinacy for absolutely continuous probability distributions when 0<q<1.

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

The paper shows that for 0

Core claim

Given 0<q<1, every absolutely continuous distribution can be described in terms of a probability density function and also in terms of a q-density. Correspondingly, it has a sequence of moments and a sequence of q-moments if those exist. New conditions on the q-moment determinacy of probability distributions are derived, along with results comparing the moment and q-moment determinacy properties.

What carries the argument

The q-density, which defines the q-moments for an absolutely continuous distribution.

If this is right

  • Distributions may be q-moment determinate without being moment determinate.
  • Comparisons reveal relations between the two forms of determinacy.
  • New sufficient conditions ensure uniqueness via q-moments.

Where Pith is reading between the lines

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

  • The approach could be tested on heavy-tailed distributions where moments diverge but q-moments converge.
  • Further work might extend the comparison to non-absolutely continuous cases.

Load-bearing premise

The distribution is absolutely continuous, admits a q-density, and the q-moments exist.

What would settle it

Two different absolutely continuous distributions sharing identical q-moments while meeting the new conditions would disprove the determinacy claim.

read the original abstract

Given $0<q<1,$ every absolutely continuous distribution can be described in two different ways: in terms of a probability density function and also in terms of a $q$-density. Correspondingly, it has a sequence of moments and a sequence of $q$-moments if those exist. In this article, new conditions on the $q$-moment determinacy of probability distributions are derived. In addition, results related to the comparison of the properties of probability distributions with respect to the moment and $q$-moment determinacy are presented.

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

2 major / 1 minor

Summary. The paper claims that for 0<q<1 every absolutely continuous probability distribution admits both an ordinary density and a q-density (hence both ordinary moments and q-moments when they exist), derives new conditions guaranteeing q-moment determinacy, and presents comparisons between the ordinary-moment and q-moment determinacy properties of distributions.

Significance. If the q-density construction and the ensuing determinacy criteria can be placed on a rigorously qualified footing, the work would supply concrete new tools for the q-analogue of the classical moment problem and for comparing uniqueness properties across the two moment sequences. No machine-checked proofs or parameter-free derivations are present.

major comments (2)
  1. [Abstract] Abstract (and presumably §1): the universal assertion that 'every absolutely continuous distribution can be described ... in terms of a q-density' is stated without support restrictions. q-densities are constructed via q-integrals or q-exponentials, which are standardly defined only on [0,∞). The claim therefore fails for laws with negative support or two-sided unbounded support; this assumption is load-bearing for every subsequent determinacy condition and comparison result.
  2. [Introduction / main theorems] The weakest_assumption paragraph already flags the missing technical conditions on support and tail behavior. Because the paper supplies no counter-examples, boundary-case analysis, or explicit statement of the support hypothesis under which the q-density exists, it is impossible to verify whether the derived conditions are correctly scoped or whether they tacitly restrict to non-negative random variables.
minor comments (1)
  1. Notation for the q-density and the q-integral should be introduced with a short self-contained definition or reference to the precise q-calculus conventions employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify support assumptions. We agree that the universal claim requires qualification and will revise the manuscript to make the non-negative support hypothesis explicit throughout.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and presumably §1): the universal assertion that 'every absolutely continuous distribution can be described ... in terms of a q-density' is stated without support restrictions. q-densities are constructed via q-integrals or q-exponentials, which are standardly defined only on [0,∞). The claim therefore fails for laws with negative support or two-sided unbounded support; this assumption is load-bearing for every subsequent determinacy condition and comparison result.

    Authors: We agree that the q-density is standardly defined via q-integrals on [0,∞) and that the abstract claim is too broad. The manuscript implicitly restricts to non-negative random variables (as is conventional in q-calculus), but this was not stated. We will revise the abstract, §1, and all theorem statements to explicitly require support in [0,∞) and note that the results do not apply to distributions with negative mass. revision: yes

  2. Referee: [Introduction / main theorems] The weakest_assumption paragraph already flags the missing technical conditions on support and tail behavior. Because the paper supplies no counter-examples, boundary-case analysis, or explicit statement of the support hypothesis under which the q-density exists, it is impossible to verify whether the derived conditions are correctly scoped or whether they tacitly restrict to non-negative random variables.

    Authors: The weakest_assumption paragraph does mention related issues but does not give a precise hypothesis. We will expand it into an explicit 'Assumptions' subsection stating that all distributions are supported on [0,∞), that the q-density is constructed via the q-integral on that interval, and that tail conditions for q-moment existence are as in the classical case. No counter-examples for negative support will be added, as they lie outside the revised scope. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations are direct from q-density and q-moment definitions

full rationale

The paper states that every absolutely continuous distribution has both an ordinary density and a q-density (0<q<1), then derives new conditions on q-moment determinacy and comparisons to ordinary moment determinacy. No quoted step reduces a claimed result to a fitted input, self-definition, or load-bearing self-citation; the abstract and structure indicate standard mathematical derivation of sufficient conditions from the given objects without renaming known results or smuggling ansatzes. The work is self-contained against external benchmarks in probability theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the work appears to rest on standard properties of q-densities and moment sequences from prior q-calculus literature.

pith-pipeline@v0.9.0 · 5606 in / 964 out tokens · 13558 ms · 2026-05-24T23:43:41.159309+00:00 · methodology

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Reference graph

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12 extracted references · 12 canonical work pages

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