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The density of a noncentral gamma difference is one-sided log-convex exactly when the positive summand is central with shape at most one.

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2026-07-13 02:34 UTC pith:PO4I7T2C

load-bearing objection Clean, elementary if-and-only-if shape classifications for noncentral gamma differences and sums; the product-of-normals corollary is the real novelty, and the proofs hold up.

arxiv 2607.09499 v1 pith:PO4I7T2C submitted 2026-07-10 math.PR math.CAmath.STstat.TH

Log-convexity and log-concavity of noncentral gamma sums and differences

Robert E. Gaunt , Fr\'ed\'eric Ouimet This is my paper
classification math.PR math.CAmath.STstat.TH MSC 60E0526A5162H10
keywords gamma distributionlog-concave densitylog-convex densityMcKay Type I distributionnoncentral chi-square distributionproduct of correlated normal random variablesvariance-gamma distribution
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper classifies when the density of a sum or difference of two independent noncentral gamma random variables is log-convex or log-concave. For the difference D = G1 - G2, the density is log-convex on the positive half-line if and only if G1 is central and has shape at most one, and symmetrically on the negative half-line; when both shapes are at least one the density is log-concave on the whole line, and in the central case those shape conditions are also necessary. For the sum the paper gives a complete log-convexity classification in the central case (shape sum at most one), log-concavity whenever both shapes are at least one, and complete classifications when the scales coincide. The same statements immediately yield one-sided log-convexity criteria for the product of two correlated normals with arbitrary means and variances, and matching shape classifications for the variance-gamma and McKay Type I densities. Readers who use these distributions in reliability, queuing, or finance now have sharp, checkable conditions that decide whether the densities preserve log-convexity or log-concavity under the usual operations of convolution and reflection.

Core claim

The density of the noncentral gamma difference D = G1 - G2 is log-convex on (0, ∞) if and only if λ1 = 0 and a1 ≤ 1 (and symmetrically on (-∞, 0)); when a1, a2 ≥ 1 the density is log-concave on R, and these conditions are necessary in the central case. Parallel complete classifications hold for central sums and for common-scale sums, with immediate corollaries for the product of correlated normals, variance-gamma, and McKay Type I densities.

What carries the argument

A fixed-limit convolution formula for the density on each open half-line, combined with the elementary preservation of log-convexity under positive mixtures and a tail obstruction that uses the asymptotic growth of the modified Bessel function of the first kind to rule out log-convexity whenever a non-centrality parameter is positive.

Load-bearing premise

The argument that rules out log-convexity for positive non-centrality rests on being allowed to pass a Bessel asymptotic inside the convolution integral because the integrand is assumed to admit a uniform asymptotic expansion in the integration variable.

What would settle it

For any fixed parameters with λ1 > 0, compute or simulate the second derivative of log p-(x) for large positive x and check whether it stays non-negative; if it becomes negative while the claimed asymptotic still holds, the uniformity step fails and the necessity claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The product of two correlated normals is one-sided log-convex precisely when the corresponding linear combination of standardized means vanishes.
  • Variance-gamma densities are log-convex on each open half-line exactly for shape parameter ν in (-1/2, 1/2] and log-concave exactly for ν ≥ 1/2.
  • McKay Type I densities are log-convex on (0, ∞) if and only if m ≤ 0 and log-concave if and only if m ≥ 1/2.
  • Any linear combination of two independent noncentral chi-squares inherits the same one-sided log-convexity and log-concavity criteria after a simple rescaling of degrees of freedom and non-centrality.

Where Pith is reading between the lines

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

  • The same convolution-plus-tail method should decide one-sided shape for differences of more than two noncentral gammas or of other Poisson mixtures of gammas.
  • Because log-concavity is preserved under convolution, the criteria immediately supply log-concave densities for sums of several independent noncentral gammas each of shape at least one.
  • The borderline case u = 1/2 recovers the asymmetric Laplace density as the unique log-affine member of the variance-gamma family, suggesting that similar borderlines may isolate other classical exponential-tailed laws.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper classifies log-convexity and log-concavity of densities of sums and differences of two independent noncentral gamma random variables. Theorem 2.3 gives a complete one-sided log-convexity classification for the difference D = G1 - G2 (log-convex on (0, ∞) iff λ1 = 0 and a1 ≤ 1, and symmetrically on (-∞, 0)), together with log-concavity on R when both shapes are at least 1, which is also necessary in the central case. Theorem 2.4 gives the corresponding sum classifications: complete log-convexity for central sums (a1 + a2 ≤ 1), log-concavity when both shapes are at least 1, and complete log-convexity/log-concavity in the common-scale case via reduction to a single noncentral gamma. Corollaries recover one-sided log-convexity for the product of two correlated normals, and shape classifications for variance-gamma and McKay Type I densities. Proofs rely on fixed-limit convolution representations, Hölder mixtures, Prékopa preservation of log-concavity, and standard Bessel/gamma asymptotics.

Significance. The results give clean, sharp if-and-only-if shape criteria for a family that unifies several classical distributions (product of correlated normals, variance-gamma, McKay Type I). The product-normal and McKay Type I classifications appear new; the variance-gamma case recovers a known special case of generalized hyperbolic shape results. The arguments are elementary and largely self-contained (convolution kernels, classical preservation theorems, standard asymptotics), with an independent backup for the noncentral tail via the authors’ earlier expansion. This is a solid, usable contribution to the literature on log-concave/log-convex densities.

minor comments (4)
  1. In Section 3.3 the interchange of the Bessel asymptotic (2.5) inside the convolution (3.1) is justified by a brief appeal to uniform asymptotic expansions (López 1999). Remark 3.1 already notes that the same tail (3.2) follows from [13, Thm 3.4]; a one-sentence pointer at the first use of (3.2) would make the argument fully self-contained without relying on the uniformity citation.
  2. Lemma 2.1 packages three distinct facts (log-convex mixtures, the convexity obstruction for G(x)/x o 0, and Prékopa convolution preservation). Splitting them or naming the classical sources more explicitly would improve readability for readers who only need one of the three.
  3. Corollary 2.6 states the product-normal log-convexity conditions cleanly, but a short remark that the density is never log-concave on either half-line (already noted in Remark 2.1) could be moved into the corollary statement itself for completeness.
  4. Typographical consistency: the arXiv identifier in the header is 2607.09499 while some self-citations use 2605.15386; ensure final bibliographic data are aligned. Also, a few accents in author names and affiliations appear inconsistently rendered.

Circularity Check

0 steps flagged

No circularity: shape classifications rest on elementary convolution, classical preservation theorems, and standard Bessel/gamma asymptotics.

full rationale

The paper derives log-convexity/log-concavity criteria for noncentral-gamma sums and differences from the characteristic-function definition (1.1), the Poisson-mixture / modified-Bessel density (1.3), and the classical facts collected in Lemma 2.1 (Hölder for log-convex integrals, Prékopa convolution preservation, and the elementary convexity obstruction of Lemma 2.1). Sufficiency of the one-sided log-convexity statements (Theorem 2.3(i)–(ii)) follows by writing the half-line density as a fixed-limit convolution whose kernel is a log-convex power times an exponential; necessity uses only the large-x asymptotic of I_η (or the elementary power-law tail when λ=0) together with the same convexity obstruction. Log-concavity when both shapes are at least one is immediate from Prékopa. The sum statements (Theorem 2.4) are likewise elementary: the central case reduces to a Beta-type integral whose log-convexity is again controlled by Lemma 2.1, while the common-scale case collapses to a single noncentral gamma whose shape is already classified in Lemma 2.2. Self-citations supply only distributional representations (product of correlated normals as a weighted noncentral-χ^{2} difference, variance-gamma as a central-gamma difference, McKay Type I as a central-gamma sum) that are either classical or re-derived; none of the shape criteria is obtained by fitting a parameter or by renaming a prior result of the authors. The single technical soft spot (interchange of the Bessel asymptotic inside the convolution) is not load-bearing: the same tail form is available from an independent expansion, and the central claims remain elementary without it. Consequently the if-and-only-if statements stand on independent, self-contained grounds and exhibit no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper is pure mathematics. It relies only on the standard definition of the noncentral gamma law, classical convexity facts, and known asymptotic expansions of modified Bessel functions. No free parameters are fitted and no new entities are postulated.

axioms (4)
  • domain assumption Definition of noncentral gamma via characteristic function (1.1) and density formulas (1.2)–(1.3)
    Taken as the starting point; standard in the literature (Mathai 1993, Gaunt–Sutcliffe).
  • standard math Log-convexity is preserved under positive mixtures (Hölder) and log-concavity under convolution (Prékopa)
    Lemma 2.1; classical functional inequalities used throughout the proofs.
  • standard math Asymptotic expansions of I u and K u (NIST Handbook 10.25.2, 10.30.4) and uniform expansion justification (López 1999)
    Used to obtain tail asymptotics that rule out log-convexity when noncentrality is positive.
  • domain assumption Shape classification of a single noncentral gamma / noncentral chi-square density (Yu 2011)
    Lemma 2.2 imports the log-concavity half from Yu; the log-convexity half is re-proved by a tail argument.

pith-pipeline@v1.1.0-grok45 · 16448 in / 2485 out tokens · 21536 ms · 2026-07-13T02:34:28.807337+00:00 · methodology

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read the original abstract

We study log-convexity and log-concavity of densities obtained from sums and differences of two independent noncentral gamma random variables. We give a complete classification of one-sided log-convexity for noncentral gamma differences, a complete log-convexity classification for sums of two independent central gamma random variables, and sharp log-concavity criteria for central differences and for common-scale sums. As special cases, we deduce a log-convexity classification for the density of the product of two correlated normal random variables with arbitrary means and variances, and log-convexity and log-concavity classifications for the densities of the variance-gamma and McKay Type I distributions.

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