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:
where
The vector [] may be used as a representation of the sample mean and may be taken as a 2-dimensional random variate.[2] 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:
where is the bivariate normal distribution and is the covariance matrix for the circular distribution:
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.
Rice, John A. (1995). Mathematical Statistics and Data Analysis (2nd ed.). Duxbury Press.