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arxiv: 2606.18458 · v1 · pith:QO6DSA5Knew · submitted 2026-06-16 · 🧮 math.PR · math-ph· math.MP

Stable size-biasing and the positive scale-mixture order of generalized Gaussian laws

Domingos S. P. Salazar This is my paper

Pith reviewed 2026-06-26 22:32 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords generalized Gaussianscale mixturestable distributionsize-biasingMellin transformpositive definitemultiplicative cocyclecharacteristic function
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The pith

A generalized Gaussian with smaller exponent is a unique positive scale mixture of one with larger exponent, but not conversely.

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

The paper classifies when one centered generalized Gaussian X_r, with density proportional to exp(-|x|^r/2), can be obtained from another by multiplication with an independent positive random variable. It proves that X_p equals in distribution V times X_q for some such V if and only if p is at most q, and that the law of V is unique. When p is less than q an explicit construction is given by taking a power of an alpha-stable random variable, size-biasing it, and rescaling; the resulting factor satisfies a multiplicative cocycle across three parameters. This also shows that the Mellin quotient between the two laws is positive definite precisely in the p less than or equal to q direction, recovering earlier Gaussian and product cases inside one classification.

Core claim

We prove that, for p,q>0, there exists a strictly positive random variable V, independent of X_q, such that X_p ≃^d V X_q if and only if p ≤ q. Moreover, the law of V is unique. For p<q, put a=1/p, b=1/q, and α=b/a=p/q. If S_α is a positive α-stable random variable with Laplace transform E exp(-u S_α)=exp(-u^α), set W_0=S_α^{-b}, let W be the W_0-size-biased version of W_0, and define V_{p,q}=2^{a-b}W. Then X_p ≃^d V_{p,q} X_q. For p>q, the required Mellin quotient, viewed as the candidate characteristic function of log V, is unbounded by Stirling's formula, and hence cannot be a characteristic function. The factor laws form a multiplicative cocycle V_{p,r} ≃^d V_{p,q} V_{q,r} for p≤q≤r, whe

What carries the argument

The size-biased version of W_0 = S_α^{-b} (S_α positive α-stable) rescaled to V_{p,q}, which supplies the explicit mixing factor when p<q and shows the Mellin quotient cannot be a characteristic function when p>q.

If this is right

  • The factors satisfy the cocycle identity V_{p,r} equals in distribution the product of independent copies of V_{p,q} and V_{q,r} whenever p≤q≤r.
  • The Mellin quotient Φ_{p,q} is positive definite exactly when p≤q.
  • The inverse Fourier-Mellin transform produces a genuine nonnegative probability density throughout the p<q branch.
  • The known Gaussian-base case and bounded-parameter product cases are recovered as instances inside the single positive scale-mixture classification.

Where Pith is reading between the lines

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

  • The cocycle structure on the exponents suggests the parameter space carries a semigroup operation that could be checked for other one-parameter families of symmetric distributions.
  • The explicit stable-based construction supplies a simulation route from one generalized Gaussian to another that could be tested for efficiency in Monte Carlo work.
  • Similar scale-mixture orders might be sought for asymmetric or multivariate extensions of the generalized Gaussian family.

Load-bearing premise

The Mellin quotient for p greater than q is unbounded and therefore cannot be the characteristic function of log V.

What would settle it

Numerical evaluation of the Mellin quotient for concrete values with p>q to confirm it is unbounded, or Monte Carlo sampling to check whether any positive multiple of X_q can match the law of X_p.

read the original abstract

Let $X_r\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\exp(-|x|^r/2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_p\stackrel{d}{=}VX_q$ if and only if $p\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1/p$, $b=1/q$, and $\alpha=b/a=p/q$. If $S_\alpha$ is a positive $\alpha$-stable random variable with Laplace transform $\mathbb{E}\exp(-uS_\alpha)=\exp(-u^\alpha)$, set $W_0=S_\alpha^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and define $V_{p,q}=2^{a-b}W$. Then $X_p\stackrel{d}{=}V_{p,q}X_q$. For $p>q$, the required Mellin quotient, viewed as the candidate characteristic function of $\log V$, is unbounded by Stirling's formula, and hence cannot be a characteristic function. The factor laws form a multiplicative cocycle, $V_{p,r}\stackrel{d}{=}V_{p,q}V_{q,r}$, for $p\le q\le r$, where the factors on the right-hand side are independent copies. Thus the Mellin quotient isolated by Dytso, Bustin, Poor and Shamai is realized constructively throughout the $p<q$ branch. In particular, $\Phi_{p,q}$ is positive definite exactly in the range $p\le q$, and the inverse Fourier--Mellin candidate density in the remaining $p<q$ branch is a genuine nonnegative probability density. The known Gaussian-base and bounded-parameter product cases are recovered as parts of a single positive scale-mixture classification.

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

0 major / 3 minor

Summary. The manuscript proves that for centered unit-scale generalized Gaussian random variables X_r (density proportional to exp(-|x|^r/2)), there exists a strictly positive V independent of X_q such that X_p equals V X_q in distribution if and only if p ≤ q, with the law of V unique. For p < q an explicit construction is given via size-biasing of W_0 = S_α^{-b} (α = p/q, S_α positive α-stable) to obtain V_{p,q} = 2^{a-b} W; for p > q the Mellin quotient, continued to the imaginary line, is shown unbounded by Stirling and hence cannot be a characteristic function. The factors satisfy the multiplicative cocycle V_{p,r} =^d V_{p,q} V_{q,r} (independent copies) for p ≤ q ≤ r, realizing the Mellin quotient of Dytso et al. constructively and establishing positive-definiteness exactly on p ≤ q.

Significance. If the derivations hold, the result supplies a complete classification of the positive scale-mixture order on generalized Gaussians, with an explicit stable size-biasing construction that realizes the Mellin quotient as a genuine characteristic function precisely when p ≤ q. The cocycle property and recovery of the Gaussian-base and bounded-parameter product cases as special instances of a single framework are notable strengths; uniqueness follows immediately from uniqueness of characteristic functions once the Mellin transform matches.

minor comments (3)
  1. Abstract: the phrase 'the W_0-size-biased version of W_0' is standard but would be clearer if the explicit Radon-Nikodym factor (or the resulting density of W) were recalled in one sentence.
  2. The normalization constant implicit in the density of X_r is omitted throughout; while conventional, a single sentence recalling that the constant is (r/2)^{1/r} / (2 Γ(1/r)) would aid readers outside the immediate area.
  3. The statement that 'the inverse Fourier-Mellin candidate density ... is a genuine nonnegative probability density' for p < q would benefit from an explicit reference to the inversion formula used (e.g., the Bromwich contour or Fourier inversion on the log scale).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are pleased that the referee finds the classification of the positive scale-mixture order, the explicit stable size-biasing construction, the cocycle property, and the recovery of prior special cases to be strengths of the work. The recommendation to accept is appreciated.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central result is an if-and-only-if characterization proved in two directions. For p < q the forward direction is established by an explicit construction: V is obtained from the size-bias of a power of an α-stable random variable whose Laplace transform is the standard one-parameter family; the resulting Mellin transform is then verified by direct computation to equal the Gamma-function ratio of moments of the generalized Gaussians. For p > q the converse is ruled out by applying the classical vertical-strip Stirling asymptotics to show that the same Gamma ratio, continued to the imaginary line, is unbounded and therefore cannot be a characteristic function. Both steps rely on externally known analytic facts (stable-law Laplace transforms, Stirling's formula, uniqueness of characteristic functions) rather than on any fitted parameter, self-referential definition, or load-bearing self-citation. The cocycle property and positive-definiteness statements are immediate consequences of the same explicit construction. No step reduces the claimed theorem to a tautology or to a renaming of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The work rests on standard properties of positive stable laws, Mellin transforms, and Stirling's approximation; no free parameters or new entities are introduced.

axioms (3)
  • standard math Positive α-stable random variables exist with the stated Laplace transform E[exp(-u S_α)] = exp(-u^α)
    Invoked to construct W_0 = S_α^{-b} for the scale factor when p < q
  • standard math Size-biasing preserves the required positivity and independence properties
    Used to obtain W from W_0
  • standard math Stirling's formula gives the asymptotic growth of the Mellin quotient
    Applied to show the candidate characteristic function is unbounded when p > q

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