An impossibility result for process discrimination

Two series of binary observations x₁; x₁,... and y₁, y₂,... are presented: at each time n isin Nopf we are given x_n and y_n. It is assumed that the sequences are generated independently of each other by two stochastic processes. We are interested in the question of whether the sequences represent a typical realization of two different processes or of the same one. We demonstrate that this is impossible to decide in the case when the processes are B-processes. It follows that discrimination is impossible for the set of all (finite-valued) stationary ergodic processes in general. This result means that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some of such processes. It contrasts earlier positive results on B-processes, in particular those showing that there are consistent dmacr-distance estimates for this class of processes.