A Constructive Approach to $q$-Gaussian Distributions: $\alpha$-Divergence as Rate Function and Generalized de Moivre-Laplace Theorem

Antonio M. Scarfone; Hiroki Suyari

arxiv: 2603.21391 · v2 · submitted 2026-03-22 · 🧮 math.PR · cond-mat.stat-mech· cs.IT· math.IT

A Constructive Approach to q-Gaussian Distributions: α-Divergence as Rate Function and Generalized de Moivre-Laplace Theorem

Hiroki Suyari , Antonio M. Scarfone This is my paper

Pith reviewed 2026-05-15 00:52 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.stat-mechcs.ITmath.IT

keywords q-Gaussian distributionsα-divergencelarge deviation principlegeneralized binomial distributionde Moivre-Laplace theoremnonlinear differential equationpower-law distributionsinformation geometry

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The pith

A generalized binomial distribution constructed from the nonlinear equation dy/dx = y^q obeys the large deviation principle with α-divergence rate function for 0 < q < 1 and converges to the q-Gaussian under n^{q/2} scaling.

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

The paper starts from the nonlinear differential equation dy/dx = y^q and develops algebraic and combinatorial rules that produce a generalized binomial distribution based on finite counting, without presupposing any particular probability law. For values of q between 0 and 1 this distribution satisfies a large deviation principle whose rate function is the α-divergence, thereby linking the counting construction directly to information geometry. The authors further prove a generalized de Moivre-Laplace theorem establishing that the same distribution converges to the heavy-tailed q-Gaussian limit, with the required scaling of fluctuations growing as n to the power q/2 because of the underlying nonlinearity. The macroscopic scaling fails for q greater than 1, and the overall framework is shown to unify the usual shift-invariant exponential family with the rescaling-invariant power-law family. A reader would care because the construction supplies an explicit counting-process origin for power-law tails that mirrors the classical binomial route to the Gaussian.

Core claim

Starting from the nonlinear differential equation dy/dx = y^q without assuming a specific distribution a priori, the authors build the algebraic and combinatorial foundations that produce a generalized binomial distribution based on finite counting. They prove the large deviation principle for this distribution in the regime 0 < q < 1, identifying the α-divergence as the rate function, and show that the macroscopic scaling breaks down for heavier tails when q > 1. They also prove a generalized de Moivre-Laplace theorem establishing convergence of the generalized binomial to the q-Gaussian distribution, with the scaling law of order n^{q/2} following from the nonlinearity. The results are num

What carries the argument

The generalized binomial distribution obtained by turning the nonlinear differential equation dy/dx = y^q into algebraic and combinatorial counting rules for a finite process.

If this is right

The large deviation principle with α-divergence rate function holds exactly when 0 < q < 1.
The generalized binomial converges in the scaled limit to the q-Gaussian distribution.
The fluctuation scaling is of order n^{q/2} rather than the classical square-root scaling.
The same macroscopic scaling fails for q > 1, where the heavy-tail regime changes the large-deviation structure.
The construction places the power-law family on the same footing as the classical exponential family.

Where Pith is reading between the lines

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

The same differential-equation starting point could generate other heavy-tailed limits by replacing the power q with a different nonlinearity.
The emergence of α-divergence from a counting process suggests that information-geometric quantities may appear generically in scaled combinatorial statistics.
Models in physics or finance that already employ q-Gaussians could now be equipped with an explicit binomial-like generating mechanism for simulation and inference.
The breakdown at q = 1 may mark a phase-transition boundary between light-tailed and heavy-tailed large-deviation regimes in nonlinear counting models.

Load-bearing premise

The nonlinear differential equation dy/dx = y^q can be converted into a finite counting process whose probabilities remain well-defined and satisfy the exact scaling needed for the large deviation principle to hold.

What would settle it

Direct Monte Carlo sampling of the generalized binomial counts for a fixed q in (0,1) whose empirical large-deviation rate function deviates from the explicit α-divergence formula by more than sampling error.

read the original abstract

The Large Deviation Principle (LDP) and the Central Limit Theorem (CLT) are central pillars of probability theory. While their formulations are established under the i.i.d. assumption, the probabilistic foundation for power-law distributions has primarily evolved through descriptive models or variational principles, rather than a constructive derivation comparable to the classical binomial process. This paper establishes a constructive probabilistic framework for power-law distributions, proceeding from the nonlinear differential equation $dy/dx = y^q$ without assuming a specific distribution a priori. We build the algebraic and combinatorial foundations, which lead to a generalized binomial distribution based on finite counting. We prove the LDP for this generalized binomial distribution in the regime $0 < q < 1$, demonstrating that the $\alpha$-divergence is identified as the rate function, and clarify the breakdown of this macroscopic scaling for heavier tails ($q > 1$). This result connects our constructive framework to the structures of information geometry. Furthermore, we prove a generalized de Moivre-Laplace theorem, showing that the generalized binomial distribution converges to a heavy-tailed limit distribution (the $q$-Gaussian distribution). We derive that the scaling law follows the order of $n^{q/2}$ as a consequence of the underlying nonlinearity. These analytical results are numerically verified for distinct values of $q \in (0, 2)$. This framework provides a constructive basis that unifies the shift-invariant exponential family and the rescaling-invariant power-law family.

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 builds a generalized binomial from dy/dx = y^q and claims LDP with alpha-divergence rate plus convergence to q-Gaussian, but the combinatorial step from DE to counting measure stays too vague to confirm the claims are fully derived.

read the letter

The main takeaway is that this work starts from the nonlinear DE dy/dx = y^q, constructs a generalized binomial via algebraic and combinatorial steps without presupposing the distribution, and then proves an LDP for 0 < q < 1 where the rate function is the alpha-divergence, along with a generalized de Moivre-Laplace result showing convergence to the q-Gaussian under n^{q/2} scaling. They also note the breakdown for q > 1 and back the claims with numerical checks across q in (0,2). That constructive route from a simple DE to both the LDP and the heavy-tailed limit is the genuinely new piece relative to prior q-Gaussian literature, and the link to information geometry structures is a clean addition. The numerics help make the scaling and breakdown concrete. The soft spot is exactly where the stress-test flagged: the transition from the DE to an explicit finite-n counting process on sequences is not spelled out with enough detail to see how the q-deformed operations become probabilities without inserting extra normalization or regularity that might pre-determine the rate function or the scaling exponent. If those steps are clean and the LDP identification follows directly, the central claims hold; if any fitting choice is hidden there, the result becomes partly circular. The paper is aimed at researchers in probability and statistical mechanics who want probabilistic foundations for power laws rather than variational or post-hoc models. It shows honest engagement with the literature and tries to derive rather than assume. I would send it to peer review so the combinatorial details can be checked by people who work on deformed algebras and large deviations.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a constructive framework for q-Gaussian distributions starting from the nonlinear ODE dy/dx = y^q. It defines a generalized binomial distribution via algebraic and combinatorial constructions on finite counting processes, proves an LDP for 0 < q < 1 in which the α-divergence is the rate function, notes the breakdown of this scaling for q > 1, and establishes a generalized de Moivre-Laplace theorem with convergence to the q-Gaussian under n^{q/2} scaling. Numerical checks for q ∈ (0,2) are included, and the work is framed as unifying shift-invariant exponential families with rescaling-invariant power-law families.

Significance. If the construction is fully rigorous and the LDP identification holds without circularity or hidden normalization, the paper would supply a valuable constructive counterpart to the classical binomial process for heavy-tailed limits. This could strengthen links between information geometry and generalized limit theorems, offering a probabilistic foundation for power-law models that is currently missing from the literature.

major comments (3)

[Construction section] Construction section (immediately after introduction of dy/dx = y^q): the mapping from the nonlinear DE to an explicit probability measure on n-step sequences is underspecified. The combinatorial realization of q-deformed addition/product rules and the precise normalization at finite n are not given in sufficient detail to confirm that the resulting counting measure yields an LDP whose rate is exactly the α-divergence rather than being enforced by construction.
[LDP theorem] LDP theorem (the result asserted for 0 < q < 1): no derivation steps, error bounds, or explicit large-deviation upper/lower bounds are supplied in the available text. The identification of α-divergence as rate function occurs after the generalized binomial is defined; a self-contained proof is required to rule out tautology.
[Generalized de Moivre-Laplace theorem] Generalized de Moivre-Laplace theorem: the claimed n^{q/2} scaling and convergence to the q-Gaussian are asserted as consequences of the underlying nonlinearity, but the proof must address moment conditions, tail behavior, and the precise sense of convergence (e.g., in distribution or in probability) for the full range 0 < q < 2.

minor comments (2)

Notation for the α-divergence and its parameter relation to q should be stated explicitly at first use, including any q-dependent normalization constants.
Numerical verification figures should include explicit labels for each q value, sample size n, and the fitted scaling exponent to allow direct comparison with the claimed n^{q/2} law.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments that highlight areas where additional detail will strengthen the manuscript. We address each major comment below and will incorporate the requested expansions in the revised version.

read point-by-point responses

Referee: [Construction section] Construction section (immediately after introduction of dy/dx = y^q): the mapping from the nonlinear DE to an explicit probability measure on n-step sequences is underspecified. The combinatorial realization of q-deformed addition/product rules and the precise normalization at finite n are not given in sufficient detail to confirm that the resulting counting measure yields an LDP whose rate is exactly the α-divergence rather than being enforced by construction.

Authors: We agree that the construction section would benefit from greater explicitness. In the revision we will expand this section to define the q-deformed addition and product rules on finite sequences in full combinatorial detail, specify the exact counting measure on n-step paths, and derive the normalization constant directly from the ODE without presupposing large-deviation properties. This will make transparent that the probability measure is constructed first and the α-divergence rate emerges subsequently. revision: yes
Referee: [LDP theorem] LDP theorem (the result asserted for 0 < q < 1): no derivation steps, error bounds, or explicit large-deviation upper/lower bounds are supplied in the available text. The identification of α-divergence as rate function occurs after the generalized binomial is defined; a self-contained proof is required to rule out tautology.

Authors: The current text states the LDP result but omits the intermediate steps. We will insert a complete self-contained proof that begins from the explicit finite-n measure, computes the scaled cumulant generating function, applies the Gärtner-Ellis theorem, and verifies that the resulting rate function coincides with the α-divergence. Upper and lower bounds together with the necessary error estimates will be supplied to eliminate any appearance of circularity. revision: yes
Referee: [Generalized de Moivre-Laplace theorem] Generalized de Moivre-Laplace theorem: the claimed n^{q/2} scaling and convergence to the q-Gaussian are asserted as consequences of the underlying nonlinearity, but the proof must address moment conditions, tail behavior, and the precise sense of convergence (e.g., in distribution or in probability) for the full range 0 < q < 2.

Authors: We will augment the proof to treat the full interval 0 < q < 2. The n^{q/2} scaling follows from the homogeneity degree of the nonlinear ODE; moment conditions will be stated explicitly (finite moments of order less than 2/(q-1) for q > 1), tail decay will be derived from the q-Gaussian form, and convergence in distribution will be established via characteristic functions. The existing numerical checks for q ∈ (0,2) will be retained as supporting evidence. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the generalized binomial distribution from the nonlinear DE dy/dx = y^q through explicit algebraic and combinatorial constructions on finite counting processes, without presupposing the target distribution or its rate function. It then proves the LDP result identifying α-divergence as the rate function and the generalized de Moivre-Laplace convergence to the q-Gaussian under n^{q/2} scaling. These steps are presented as independent derivations with numerical verification; no equation reduces by construction to a prior definition or self-citation, and the central claims rest on the combinatorial construction rather than tautological identification.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The framework rests on the existence of a finite counting process generated by the DE dy/dx = y^q, standard LDP definitions, and the identification of α-divergence without showing that this identification is independent of the construction choices.

free parameters (1)

q
The nonlinearity parameter q is introduced via the DE and treated as given; its value controls both the regime of validity (0<q<1) and the scaling exponent q/2.

axioms (2)

domain assumption The nonlinear DE dy/dx = y^q admits a well-defined algebraic and combinatorial lift to a finite counting process (generalized binomial).
Invoked at the start of the constructive framework; no derivation of the lift is supplied in the abstract.
standard math Standard large-deviation theory applies directly to the generalized binomial once constructed.
Used to identify the rate function with α-divergence.

invented entities (1)

generalized binomial distribution no independent evidence
purpose: Finite counting process whose probabilities follow from the nonlinear DE
New object introduced to carry the constructive derivation; independent evidence would be an explicit probability mass function or generating function not reducible to known q-binomials.

pith-pipeline@v0.9.0 · 5587 in / 1684 out tokens · 24245 ms · 2026-05-15T00:52:33.008363+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 echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We prove the LDP for this generalized binomial distribution in the regime 0 < q < 1, demonstrating that the α-divergence is identified as the rate function... ln_q b_q(k;n,r) = -n^{2-q}/(2-q) D_{2-q}(p∥r) + ... where q=1-α/2
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff refines

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

Definition 10 (q-binomial distribution) ... from refined q-Stirling’s formula and q-product

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