Directional statistics is the subdiscipline of statistics that deals with directions (unit vectors in Rn), axes (lines through the origin in Rn) or rotations in Rn. The means and variances of directional quantities are all finite, so that the central limit theorem may be applied to the particular case of directional statistics.
This article will deal only with unit vectors in 2-dimensional space (R2) but the method described can be extended to the general case.
A sample of angles are measured, and since they are indefinite to within a factor of , the complex definite quantity is used as the random variate. The probability distribution from which the sample is drawn may be characterized by its moments, which may be expressed in Cartesian and polar form:
It follows that:
Sample moments for N trials are:
The vector  may be used as a representation of the sample mean and may be taken as a 2-dimensional random variate. The bivariate central limit theorem states that the joint probability distribution for and in the limit of a large number of samples is given by:
Note that the bivariate normal distribution is defined over the entire plane, while the mean is confined to be in the unit ball (on or inside the unit circle). This means that the integral of the limiting (bivariate normal) distribution over the unit ball will not be equal to unity, but rather approach unity as N approaches infinity.
It is desired to state the limiting bivariate distribution in terms of the moments of the distribution.