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：本文针对一般设置下的自随机数生成（SRNG）问题进行了研究。已有文献探讨了基于变分距离的最优SRNG率。本文首先尝试刻画基于$f$-散度子类的最优SRNG率。所考虑的$f$-散度子类涵盖典型距离度量，如变分距离、KL散度、Hellinger距离等。因此，本文结果可视为对先前基于变分距离研究结论的推广。其次，我们从多个角度分析了所得最优SRNG率。$\varepsilon$-信源编码问题是与SRNG问题密切相关的课题之一。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。进一步，我们将研究成果应用于率失真感知（RDP）函数，从而基于发现建立了关于$f$-散度的RDP函数下界。最后，我们探讨了利用光滑Rényi熵表示最优SRNG率的方法。