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 $\mu$ 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 $\mu$ 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.

中文翻译：

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Rényi散度的泛化存在的条件

我们给出了存在的Rényi散度泛化的必要和充分条件，该泛化是根据变形的指数函数定义的。如果基础措施$\mu$由于是非原子的，我们发现并非所有变形的指数函数都可以用于Rényi发散的推广。提供了涉及变形指数函数的条件。在这种情况下$\mu$ 是纯粹的原子（对自然数集的计数度量），我们证明了可以在概化中使用任何变形的指数函数。