# 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 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,...,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,∞) with a non-increasing 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.

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