Some Results on Distributed Source Simulation with no Communication

We consider the problem of distributed source simulation with no communication, in which Alice and Bob observe sequences Un and Vn respectively, drawn from a joint distribution $p_{UV}^{\otimes n}$, and wish to locally generate sequences Xn and Yn respectively with a joint distribution that is close (in KL divergence) to $p_{XY}^{\otimes n}$. We provide a single-letter condition under which such a simulation is asymptotically possible with a vanishing KL divergence. Our condition is nontrivial only in the case where the Gàcs-Körner (GK) common information between U and V is nonzero, and we conjecture that only scalar Markov chains $X-U-V-Y$ can be simulated otherwise. Motivated by this conjecture, we further examine the case where both pUV and pXY are doubly symmetric binary sources with parameters $p, q\leq 1/2$ respectively. While it is trivial that in this case $p\leq q$ is both necessary and sufficient, we show that when p is close to q then any successful simulation is close to being scalar in the total variation sense.