Characterization of multivariate distributions through a functional equation of their characteristic functions

Abstract

The multivariate distributions whose characteristic functions satisfy a given integrated functional equation are proved to be essentially multivariate stable (semi-stable) distributions. This generalizes the characterization of univariate distributions in Ramachandran and Rao (1970), Shimizu (1968, 1978), Davies and Shimizu (1976), and Ramachandran et al. (1988). Related characterization problems such as identically distributed, and zero regressions of linear statistics are also discussed.