Wrapped processes on circular lattices for planar directions

This paper introduces a class of stochastic processes making jumps around the circle. These circular processes are the wrapped versions of the Poisson, the negative binomial, the binomial processes and of extensions thereof obtained by compounding with a secondary frequency distribution. Their prevailing application is for modeling planar motions. These processes can be weakly stationary, can have uniform stationary distribution, can have independent or stationary circular increments. Their autocovariance functions, one-dimensional trigonometric moments and wrapped distributions are obtained. For the wrapped Poisson process, it is shown that the formula for the one-dimensional distribution can be obtained either by discrete Fourier transform or by wrapping the Poisson distribution around the circle and by simplifying it with the generalized hyperbolic function. Some numerical illustrations and comparisons are provided.