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Symbolic transfer entropy rate is equal to transfer entropy rate for bivariate finite-alphabet stationary ergodic Markov processes

Transfer entropy is a measure of the magnitude and the direction of information flow between jointly distributed stochastic processes. In recent years, its permutation analogues are considered in the literature to estimate the transfer entropy by counting the number of occurrences of orderings of values, not the values themselves. It has been suggested that the method of permutation is easy to implement, computationally low cost and robust to noise when applying to real world time series data. In this paper, we initiate a theoretical treatment of the corresponding rates. In particular, we consider the transfer entropy rate and its permutation analogue, the symbolic transfer entropy rate, and show that they are equal for any bivariate finite-alphabet stationary ergodic Markov process. This result is an illustration of the duality method introduced in [T. Haruna and K. Nakajima, Physica D 240, 1370 (2011)]. We also discuss the relationship among the transfer entropy rate, the time-delayed mutual information rate and their permutation analogues.