Closed-form linear moments of the two-dimensional angular central Gaussian distribution
Pith reviewed 2026-06-28 20:14 UTC · model grok-4.3
The pith
The mean of the two-dimensional angular central Gaussian angle is an arctangent of the parameters and its second moment is the real part of a dilogarithm.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By performing a contour integration around the branch cut of arctan z, the integrals defining the linear moments of the ACG density yield E[θ] as an arctangent expression in the parameters and E[θ²] as the real part of a dilogarithm at points determined by those parameters.
What carries the argument
Contour integration around the branch cut of arctan z to evaluate the moment integrals.
If this is right
- The moments can be evaluated analytically for any valid parameters without numerical methods.
- These closed forms allow exact calculations in applications where linear rather than circular moments of the angle are required.
- The approach provides a template for finding moments in similar angular distributions derived from Gaussians.
Where Pith is reading between the lines
- The contour method could be adapted to compute higher-order linear moments or moments of related distributions.
- Connections may exist to integrals arising in other areas of multivariate statistics or phase modeling in physics.
- Numerical verification for edge cases of the parameter space would strengthen confidence in the expressions.
Load-bearing premise
The contour integration around the branch cut of arctan z produces the desired moments for the relevant parameter values without encountering additional singularities or requiring further restrictions.
What would settle it
Computing the integral for E[θ] numerically with specific parameter values and finding it does not match the proposed arctangent formula would falsify the result.
read the original abstract
The polar-angle marginal of a centred bivariate Gaussian distribution, obtained after integrating out the radial coordinate, gives the two-dimensional angular central Gaussian (ACG) distribution of Tyler. While its trigonometric and vector-valued moments have been studied in detail, to our knowledge there are no explicit closed-form expressions for the \emph{linear} moments $\mathbf{E}[\theta]$ and $\mathbf{E}[\theta^{2}]$ on the natural domain $\theta\in\left]-\pi/2,\pi/2\right[$. Here \textit{linear} refers to the ordinary moments $\int\theta^{k}f(\theta)\,d\theta$ of the angle regarded as a real-valued variable, in contrast to the circular (trigonometric) moments $\mathbf{E}[e^{ik\theta}]$ customary in directional statistics. We provide such expressions: the mean is a simple arctangent of the parameters, while the second moment is given by the real part of a dilogarithm. The derivation, based on a contour integration around the branch cut of $\arctan z$, is elementary. These quantities naturally arise in physics, where $\theta$ is interpreted as a real-valued phase rather than a circular variable.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper derives closed-form expressions for the linear (non-circular) moments E[θ] and E[θ²] of the two-dimensional angular central Gaussian distribution on θ ∈ ]−π/2, π/2[, obtained as the polar-angle marginal of a centered bivariate Gaussian after radial integration. It claims that the mean is given by a simple arctangent of the covariance parameters and the second moment by the real part of a dilogarithm, with the derivation performed via an elementary contour integration around the branch cut of arctan z.
Significance. If the contour-integration steps are free of unaccounted singularities or branch-cut violations, the results supply explicit, non-numerical formulas for quantities that arise when the angle is interpreted as a real-valued phase rather than a circular variable, filling a documented gap relative to the existing literature on trigonometric moments of the ACG distribution.
major comments (1)
- [derivation (contour integration)] The central derivation (abstract and main text) evaluates the moments by contour integration around the branch cut of arctan z. The ACG density contains a quadratic denominator whose roots depend on the covariance parameters; the manuscript supplies neither an explicit verification that the chosen contour remains free of these roots for the full open set of positive-definite matrices nor a demonstration that no additional residues arise when a pole crosses the cut.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for identifying a point where the contour-integration argument requires additional rigor. We respond to the single major comment below and will revise the manuscript to incorporate the requested verification.
read point-by-point responses
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Referee: [derivation (contour integration)] The central derivation (abstract and main text) evaluates the moments by contour integration around the branch cut of arctan z. The ACG density contains a quadratic denominator whose roots depend on the covariance parameters; the manuscript supplies neither an explicit verification that the chosen contour remains free of these roots for the full open set of positive-definite matrices nor a demonstration that no additional residues arise when a pole crosses the cut.
Authors: We agree that the manuscript does not contain an explicit verification that the contour avoids the poles of the integrand for every positive-definite covariance matrix, nor a demonstration that poles cannot cross the branch cut. This is a legitimate gap in the presented derivation. In the revised version we will insert a short lemma immediately after the contour setup that (i) solves the quadratic denominator explicitly in terms of the covariance entries, (ii) shows that both roots lie strictly outside the closed contour encircling the branch cut of arctan z whenever the matrix is positive definite, and (iii) proves by continuity that the roots remain in the same half-planes as the parameters vary inside the positive-definite cone, so that no pole crosses the cut. The lemma will be elementary and will not alter the final closed-form expressions. revision: yes
Circularity Check
No circularity; derivation proceeds from integral definition via contour integration
full rationale
The paper starts from the explicit integral definition of the linear moments of the ACG density (obtained after radial integration of the bivariate Gaussian) and evaluates them using contour integration around the arctan branch cut. This produces closed-form expressions (arctan for the mean, Re(dilog) for the second moment) that are not equivalent by construction to any fitted parameter or input quantity. No self-citations are load-bearing, no uniqueness theorems are imported from prior author work, and no ansatz is smuggled via citation. The derivation is presented as direct and elementary from the moment integrals, consistent with the reader's assessment of score 1.0. The skeptic's concerns address possible singularities or domain restrictions (a correctness issue) but do not indicate any reduction of the claimed result to its own inputs.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard results of complex analysis on contour integration around branch cuts of the arctangent function
discussion (0)
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