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Blurred stochastic chains

Assume we have two stochastic chains taking values in a finite alphabet. These chains may be of infinite order. Assume also that these chains are coupled in such a way that given the past of both chains they have a not too large probability of differing. This is the case when we observe a chain through a noisy channel. This situation presumably also occurs in models for the brain activity when a chain of stimuli is presented to a volunteer and we observe a corresponding chain of neurophysiological recordings. The question is how these two chains are quantitatively related. Under suitable conditions, we obtain upper-bounds for the differences between the marginal conditional distributions of the two chains and between the probability of the next symbol of each chain, given the past of the past of one of them.