Distributions on the circle group

In this paper, we extend the definition of a random angle and the definition of a probability distribution of a random angle. We expand P. Levy’s researches related to wrapping the probability distributions defined on R. We determine a relation between quasi-lattice probability distributions on R and lattice probability distributions on the unit circle S. We use the Bergstrom identity for comparison of a convolution of probability distributions of random angles. We also prove an inverse formula for lattice probability distributions on S.