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

TL;DR

This paper provides non-asymptotic and asymptotic bounds for uniform random number generation from Markov chains, with efficient computation and detailed analysis of large deviation, moderate deviation, and second order regimes.

Contribution

It introduces computable bounds for random number generation from Markov chains and characterizes their asymptotic and second order behaviors.

Findings

Bounds are efficiently computable regardless of block length.

Asymptotic tightness of bounds in large and moderate deviation regimes.

Derived second order rates and variances for the problems.

Abstract

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. Further, we address the equivocation rates for these problems.