The self-random number generation (SRNG) problem is considered for general setting. In the literature, the optimum SRNG rate with respect to the variational distance has been discussed. In this paper, we first try to characterize the optimum SRNG rate with respect to a subclass of $f$-divergences. The subclass of $f$-divergences considered in this paper includes typical distance measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence our result can be considered as a generalization of the previous result with respect to the variational distance. Next, we consider the obtained optimum SRNG rate from several viewpoints. The $\varepsilon$-source coding problem is one of related problems with the SRNG problem. Our results reveal how the SRNG problem with the $f$-divergence relate to the $\varepsilon$-fixed-length source coding problem. We also apply our results to the rate distortion perception (RDP) function. As a result, we can establish a lower bound for the RDP function with respect to $f$-divergences using our findings. Finally, we discuss the representation of the optimum SRNG rate using the smooth R\'enyi entropy.

翻译：本文考虑一般设定下的自随机数生成(SRNG)问题。已有文献讨论了基于变分距离的最优SRNG速率。本文首先尝试刻画基于$f$-散度子类的最优SRNG速率。本文考虑的$f$-散度子类包括变分距离、KL散度、Hellinger距离等典型距离度量，因此我们的结果可视为对先前基于变分距离结果的推广。其次，我们从多个视角分析所得到的最优SRNG速率。$\varepsilon$-信源编码问题是与SRNG问题相关的问题之一。我们的结果揭示了基于$f$-散度的SRNG问题与$\varepsilon$-定长信源编码问题之间的关联。此外，我们将所得结果应用于率失真感知(RDP)函数，从而利用我们的发现建立了基于$f$-散度的RDP函数下界。最后，我们讨论了利用平滑Rényi熵对最优SRNG速率的表示。