Second Order Asymptotics for Random Number Generation

TL;DR

This paper investigates the second order asymptotics of random number generation between arbitrary probability distributions, extending previous work that focused on uniform distributions, and derives the optimal second order generation rate.

Contribution

It provides the first analysis of second order asymptotics for random number generation between arbitrary distributions, including the derivation of the optimal second order rate.

Findings

Derived the optimal second order generation rate for arbitrary distributions.

Extended second order asymptotic theory to non-uniform distributions.

Provided a comprehensive analysis for the second order resolvability problem.

Abstract

We treat a random number generation from an i.i.d. probability distribution of $P$ to that of $Q$. When $Q$ or $P$ is a uniform distribution, the problems have been well-known as the uniform random number generation and the resolvability problem respectively, and analyzed not only in the context of the first order asymptotic theory but also that in the second asymptotic theory. On the other hand, when both $P$ and $Q$ are not a uniform distribution, the second order asymptotics has not been treated. In this paper, we focus on the second order asymptotics of a random number generation for arbitrary probability distributions $P$ and $Q$ on a finite set. In particular, we derive the optimal second order generation rate under an arbitrary permissible confidence coefficient.