Reciprocal symmetry, unimodality and Khintchine’s theorem

Abstract

The symmetric distributions on the real line and their multi-variate extensions play a central role in statistical theory and many of its applications. Furthermore, data in practice often consist of non-negative measurements. Reciprocally symmetric distributions defined on the positive real line may be considered analogous to symmetric distributions on the real line. Hence, it is useful to investigate reciprocal symmetry in general, and Mudholkar and Wang’s notion of R-symmetry in particular. In this paper, we shall explore a number of interesting results and interplays involving reciprocal symmetry, unimodality and Khintchine’s theorem with particular emphasis on R-symmetry. They bear on the important practical analogies between the Gaussian and inverse Gaussian distributions.