Closed-form linear moments of the two-dimensional angular central Gaussian distribution

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 $\mathbf{E}[θ]$ and $\mathbf{E}[θ^{2}]$ on the natural domain $θ\in\left]-π/2,π/2\right[$. Here \textit{linear} refers to the ordinary moments $\intθ^{k}f(θ)\,dθ$ of the angle regarded as a real-valued variable, in contrast to the circular (trigonometric) moments $\mathbf{E}[e^{ikθ}]$ 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 $θ$ is interpreted as a real-valued phase rather than a circular variable.