The problem of exactly generating a general random process (target process) by using another general random process (coin process) is studied. The performance of the interval algorithm, introduced by Han and Hoshi, is analyzed from the perspective of information spectrum approach. When either the coin process or the target process has one point spectrum, the asymptotic optimality of the interval algorithm among any random number generation algorithms is proved, which demonstrates utility of the interval algorithm beyond the ergodic process. Furthermore, the feasibility condition of exact random number generation is also elucidated. Finally, the obtained general results are illustrated by the case of generating a Markov process from another Markov process.