Distributed Source Simulation with No Communication

We consider the problem of distributed source simulation with no communication, in which Alice and Bob observe sequences U{n} and V{n} respectively, drawn from a joint distribution p {UV} {otimes n} , and wish to locally generate sequences X{n} and Y{n} 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 p {UV} and p {XY} 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 use Fourier analytic tools to show that when p is close to q then any successful simulation is close to being scalar in the total variation sense.