# Source Coding for Synthesizing Correlated Randomness

We consider a scenario wherein two parties Alice and Bob are provided X_1^n and X_2^n samples that are IID from a PMF p_X_1 X_2. Alice and Bob can communicate to Charles over (noiseless) communication links of rate R_1 and R_2 respectively. Their goal is to enable Charles generate samples Y^n such that the triple (X_1^n,X_2^n,Y^n) has a PMF that is close, in total variation, to ∏ p_X_1 X_2 Y. In addition, the three parties may posses shared common randomness at rate C. We address the problem of characterizing the set of rate triples (R_1,R_2,C) for which the above goal can be accomplished. We provide a set of sufficient conditions, i.e., an achievable rate region for this three party setup. Our work also provides a complete characterization of a point-to-point setup wherein Bob is absent and Charles is provided with side-information.

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