An impossibility result for process discrimination

Two series of binary observations x<inf>1</inf>; x<inf>1</inf>,… and y<inf>1</inf>, y<inf>2</inf>,… are presented: at each time n ∈ ℕ we are given x<inf>n</inf> and y<inf>n</inf>. 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 d̅-distance estimates for this class of processes.

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