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arxiv: 2605.15386 · v1 · pith:5Y47J5LRnew · submitted 2026-05-14 · 🧮 math.PR

The non-central gamma sum and difference distributions: exact distribution and asymptotic expansions

Robert E. Gaunt , Heather L. Sutcliffe This is my paper

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classification 🧮 math.PR
keywords non-central gamma distributionsum distributiondifference distributionasymptotic expansionsprobability density functiontail probabilitiesquantile functions
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The pith

Exact series and integral formulas are derived for the densities of sums and differences of independent non-central gamma random variables.

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

The paper derives exact expressions for the probability densities of the sum and difference of two independent non-central gamma random variables. It presents both series and integral representations for these densities. These exact formulas are then used to derive asymptotic expansions for the densities, tail probabilities, and quantile functions. A special case yields closed-form coefficients for the asymptotic expansion of the density of the product of two correlated normal variables. Numerical results assess how well the approximations work across different parameter values.

Core claim

Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented. These formulas are then applied to obtain asymptotic expansions for the probability density function, tail probabilities and quantile functions of these distributions. As a special case, we deduce asymptotic expansions for the probability density function of the product of correlated normal random variables with the coefficients given in closed-form.

What carries the argument

Series and integral representations for the density of the sum or difference obtained via the independence of the two non-central gamma random variables.

If this is right

  • Asymptotic expansions follow for the PDF, tail probabilities, and quantile functions of the sum and difference distributions.
  • Closed-form coefficients are obtained for the asymptotic density of the product of correlated normal random variables.
  • The approximations can be checked numerically for accuracy over ranges of the distribution parameters.

Where Pith is reading between the lines

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

  • The exact representations may support further derivations such as moment calculations for these sum and difference laws.
  • The approach could extend to deriving distributions for linear combinations involving additional non-central gamma variables.

Load-bearing premise

The two random variables are independent and each follows a non-central gamma distribution, with the series and integral representations converging for the given parameter values.

What would settle it

Direct numerical evaluation or Monte Carlo simulation of the sum or difference density for specific shape and non-centrality parameters that disagrees with the derived series or integral formula beyond numerical error.

read the original abstract

Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented. These formulas are then applied to obtain asymptotic expansions for the probability density function, tail probabilities and quantile functions of these distributions. As a special case, we deduce asymptotic expansions for the probability density function of the product of correlated normal random variables with the coefficients given in closed-form. Numerical results are presented to assess the accuracy of our asymptotic approximations across a range of parameter constellations.

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

1 major / 2 minor

Summary. The manuscript derives exact series and integral representations for the probability density functions of the sum and difference of two independent non-central gamma random variables. These are obtained via convolution of the individual densities, each expanded in series involving modified Bessel functions of the first kind. The exact formulas are then employed to derive asymptotic expansions for the densities, tail probabilities, and quantile functions. A special case yields asymptotic expansions for the density of the product of two correlated normal random variables, with coefficients in closed form. Numerical results assess the accuracy of the asymptotic approximations over a range of parameter values.

Significance. If the exact representations are placed on a rigorous footing, the results supply useful closed-form and asymptotic tools for distributions arising in non-central chi-squared statistics, wireless communications, and quadratic forms. The explicit coefficients for the correlated-normal product case and the numerical validation of the approximations constitute concrete strengths that enhance applicability in large-parameter regimes.

major comments (1)
  1. [Section deriving the exact series representation for the sum] In the derivation of the series representation for the sum density (the convolution step that produces a double sum inside the integral over the positive line), the interchange of summation and integration is performed without an explicit dominated-convergence or Weierstrass M-test argument that holds uniformly for the admissible ranges of shape, scale, and non-centrality parameters. This justification is load-bearing for the claim that the resulting single series equals the true density.
minor comments (2)
  1. [Numerical results section] The integral representations are stated but their numerical evaluation is not compared against the series forms in the reported experiments; adding such a comparison would strengthen the practical assessment.
  2. A brief table summarizing the parameter regimes (shape, scale, non-centrality) for which each representation is claimed to converge would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment of its potential utility in non-central chi-squared statistics, wireless communications, and quadratic forms. We address the single major comment below and will incorporate the requested justification in the revised version.

read point-by-point responses

  1. Referee: In the derivation of the series representation for the sum density (the convolution step that produces a double sum inside the integral over the positive line), the interchange of summation and integration is performed without an explicit dominated-convergence or Weierstrass M-test argument that holds uniformly for the admissible ranges of shape, scale, and non-centrality parameters. This justification is load-bearing for the claim that the resulting single series equals the true density.

    Authors: We agree that an explicit justification for the interchange of summation and integration is required to place the series representation on a fully rigorous footing. In the revised manuscript we will insert a dedicated paragraph immediately following the convolution step. We will apply the dominated convergence theorem by exhibiting an integrable majorant: using the standard bound |I_ν(z)| ≤ (z/2)^ν / Γ(ν+1) exp(z) for the modified Bessel function together with the explicit form of the non-central gamma densities, the absolute value of the double sum is dominated by a multiple of the product of two gamma densities (shifted by the non-centrality parameters), which is integrable over (0,∞) uniformly for all admissible shape parameters >0, scale parameters >0, and non-centrality parameters ≥0. This establishes that the interchange is valid and that the resulting single series equals the true density. The same argument applies verbatim to the difference density after the appropriate change of variables. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on standard convolution and series expansions of non-central gamma densities

full rationale

The paper derives exact series and integral representations for the densities of the sum and difference by convolving the known series expansions of the individual non-central gamma densities. These steps use standard integral representations and term-by-term integration justified under the stated parameter regimes. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the central claims. The asymptotic expansions are obtained from the exact representations via standard asymptotic analysis techniques. The derivation chain is self-contained against external benchmarks such as the known non-central gamma density and convolution formulas.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of the non-central gamma distribution and the assumption of independence between the two variables; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The two random variables are independent.
    Explicitly stated as the basis for deriving the sum and difference distributions.
  • domain assumption Each variable follows a non-central gamma distribution.
    The starting point for all subsequent exact and asymptotic results.

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    Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented.

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