Uniform Random Number Generation From Markov Chains: Non-Asymptotic and Asymptotic Analyses

In this paper, we derive non-asymptotic achievability and converse bounds on the random number generation with/without side-information. Our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length. We also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds, which implies that our bounds are asymptotically tight in those regimes. We also show the second-order rates of those problems, and derive single letter forms of the variances characterizing the second-order rates. Furthermore, we address the relative entropy rate and the modified mutual information rate for these problems.