  We consider random number conversion (RNC) through random number storage with
restricted size. We clarify the relation between the performance of RNC and the
size of storage in the framework of first- and second- order asymptotics, and
derive their rate regions. Then, we show that the results for RNC with
restricted storage recover those for conventional RNC without storage in the
limit of storage size. To treat RNC via restricted storage, we introduce a new
kind of probability distributions named generalized Rayleigh-normal
distributions. Using the generalized Rayleigh-normal distributions, we can
describe the second-order asymptotic behaviour of RNC via restricted storage in
a unified manner. As an application to quantum information theory, we analyze
LOCC conversion via entanglement storage with restricted size. Moreover, we
derive the optimal LOCC compression rate under a constraint of conversion
accuracy.
