Multiple-Output Channel Simulation and Lossy Compression of Probability Distributions

We consider a variant of the channel simulation problem with a single input and multiple outputs, where Alice observes a probability distribution P from a set of prescribed probability distributions $\mathcal{P}$, and sends a prefix-free codeword W to Bob to allow him to generate n i.i.d. random variables $X_{1}, X_{2}, \ldots, X_{n}$ which follow the distribution P. This can also be regarded as a lossy compression setting for probability distributions. This paper describes encoding schemes for three cases of $P: P$ is a distribution over positive integers, P is a continuous distribution over $[0,1]$ with a non-increasing pdf, and P is a continuous distribution over $[0, \infty)$ with a nonincreasing pdf. We show that the growth rate of the expected codeword length is sub-linear in n when a power law bound is satisfied. An application of multiple-outputs channel simulation is the compression of probability distributions. A full version of this paper is accessible at: https://arxiv.org/pdf/2105.01045.pdf