Conditions for the existence of a generalization of Rényi divergence

Abstract: We give necessary and sufficient conditions for the existence of a generalization of R\'enyi divergence, which is defined in terms of a deformed exponential function. If the underlying measure $\mu$ is non-atomic, we found that not all deformed exponential functions can be used in the generalization of R\'enyi divergence; a condition involving the deformed exponential function is provided. In the case $\mu$ 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.