Strong Sub- and Super-Gaussianity

We introduce the terms strong suband super-Gaussianity to refer to the previously introduced class of densities log-concave is x and log-convex in x respectively. We derive relationships among the various definitions of suband super-Gaussianity, and show that strong suband super-Gaussianity are related to the score function being star-shaped upward or downward with respect to the origin. We illustrate the definitions and results by extending a theorem of Benveniste, Goursat, and Ruget on uniqueness of separating local optima in ICA.