A Characterization of the Uniform Distribution on the Circle

Abstract

Geary's characterization of the normal distribution asserts that if $n \geqslant 2$ i.i.d. observations come from some distribution on the line, then the sample mean and variance are independent if and only if the observations are normally distributed. A similar characterization is established here for the uniform distribution on the circle. Given a sample of $n \geqslant 2$ i.i.d. random angles from a distribution defined by a density on the circle satisfying some mild regularity conditions, the sample mean direction and resultant length are independent if and only if the angles come from the uniform distribution.