  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.
