Sampling with arbitrary precision
We study the problem of the generation of a continuous random variable when a source of independent fair coins is available. We first motivate the choice of a natural criterion for measuring accuracy, the Wasserstein $L_\infty$ metric, and then show a universal lower bound for the expected number of required fair coins as a function of the accuracy. In the case of an absolutely continuous random variable with finite differential entropy, several algorithms are presented that match the lower bound up to a constant, which can be eliminated by generating random variables in batches.