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 $\mathbf{E}[\hat{I}¸]$ and $\mathbf{E}[\hat{I}{¸}^2]$ on the natural domain $\hat{I}¸\in \left]-\ddot{I}€/2,\ddot{I}€/2\right[$. Here \textit{linear} refers to the ordinary moments $\int \hat{I}{¸}^{k}f(\hat{I}¸)d\hat{I}¸$ of the angle regarded as a real-valued variable, in contrast to the circular (trigonometric) moments $\mathbf{E}[e^{ik\hat{I}¸}]$ 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 $\hat{I}¸$ is interpreted as a real-valued phase rather than a circular variable.

7 pages, 1 figure