Some Results on Distributed Source Simulation with no Communication

Tomer Berg, Ofer Shayevitz, Young-Han Kim, Lele Wang

Abstract: We consider the problem of distributed source simulation with no communication, in which Alice and Bob observe sequences U <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and V <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> respectively, drawn from a joint distribution $p_{UV}^{\otimes n}$, and wish to locally generate sequences X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and Y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> 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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">UV</inf> and p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">XY</inf> 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.