Abstract
This chapter focuses on the fundamental theorems and distribution theory. The chapter discusses the uniqueness theorem, inversion formulae, a limit theorem on sequences of random variables, central limit theorems on the circle, and Poincaré's theorem. The distribution theory for a sample resultant and for some related statistics is also presented in the chapter. The chapter also discusses the isotropic random walk and presents sampling distributions of various statistics from the uniform and von Mises distributions. The continuity theorem—theorem of Carathéodory—is also discussed in the chapter. The proof of the continuity theroem is much simpler for circular distributions than for distributions on the line. It is not possible for the probability mass to “escape to infinity” on the circle. The chapter reviews the distributions for a multi-modal von Mises-type population and the limiting distributions of angular statistics.
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