Conditions for the existence of a generalization of Rényi divergence

Abstract

We give necessary and sufficient conditions for the existence of a generalization of Rényi divergence, which is defined in terms of a deformed exponential function. If the underlying measure is non-atomic, we found that not all deformed exponential functions can be used in the generalization of Rényi divergence; a condition involving the deformed exponential function is provided. In the case is purely atomic (the counting measure on the set of natural numbers), we show that any deformed exponential function can be used in the generalization.