Random-step Markov processes

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We explore two notions of stationary processes. The first is called a random-step Markov process in which the stationary process of states, (Xᵢ)ᵢ ∈ Z has a stationary coupling with an independent process on the positive integers, (Lᵢ)ᵢ ∈ Z of `random look-back distances'. That is, L₀ is independent of the `past states', (Xᵢ, Lᵢ)ᵢ<₀, and for every positive integer n, the probability distribution on the `present', X₀, conditioned on the event {L₀ = n} and on the past is the same as the probability distribution on X₀ conditioned on the `n-past', (Xᵢ)₋ₙ≤ ᵢ <₀ and {L₀ = n}. A random Markov process is a generalization of a Markov chain of order n and has the property that the distribution on the present given the past can be uniformly approximated given the n-past, for n sufficiently large. Processes with the latter property are called uniform martingales, closely related to the notion of a `continuous g-function'. We show that every stationary process on a countable alphabet that is a uniform martingale and is dominated by a finite measure is also a random Markov process and that the random variables (Lᵢ)ᵢ ∈ Z and associated coupling can be chosen so that the distribution on the present given the n-past and the event {L₀ = n} is `deterministic': all probabilities are in {0,1}. In the case of finite alphabets, those random-step Markov processes for which L₀ can be chosen with finite expected value are characterized. For stationary processes on an uncountable alphabet, a stronger condition is also considered which is sufficient to imply that a process is a random Markov processes. In addition, a number of examples are given throughout to show the sharpness of the results.