An ergodic process of zero divergence-distance from its time-reversed process

An ergodic process $P$ is constructed such that the divergence-rate $D(P || P^*)$ is zero, yet $P$ is not equal to its time-reversed process $P^*$. The process $P$ is constructed as a special realization of the universal coding found by Xu. This result shows that there exist time-asymmetric processes that are indistinguishable under time-reversal and that can, in principle, be generated with zero dissipation.