Inversion of Characteristic Functions Without Imaginary Unit

We start from a pure vector update of the concepts of complex number and characteristic function of a probability distribution, free from the imaginary unit, and reprove some basic properties of the latter. In particular, we derive vector inverse formulas for probability distributions and density functions, provide a new geometric proof of the convergence of suitably centered and normalized vector powers of characteristic functions, and thus provide an update to the proof of the central limit theorem.