# Random-step Markov processes

Neal Bushaw, Karen Gunderson, Steven Kalikow

Research output: Contribution to journalArticle (Academic Journal)

## Abstract

We explore two notions of stationary processes. The first is called a random-step Markov process in which the stationary process of states, $(X_i)_{i\in \mathbb{Z}}$ has a stationary coupling with an independent process on the positive integers, $(L_i)_{i \in \mathbb{Z}}$ of random look-back distances'. That is, $L_0$ is independent of the past states', $(X_i,L_i)_{i<0}$, and for every positive integer $n$, the probability distribution on the present', $X_0$, conditioned on the event $\{L_0=n\}$ and on the past is the same as the probability distribution on $X_0$ conditioned on the $n$-past', $(X_i)_{−n \leq i<0}$ and $\{L_0=n\}$. 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 $(L_i)_{i\in \mathbb{Z}}$ and associated coupling can be chosen so that the distribution on the present given the $n$-past and the event $\{L_0=n\}$ is deterministic': all probabilities are in $\{0,1\}$. In the case of finite alphabets, those random-step Markov processes for which $L_0$ 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.
Original language English 31 arXiv Published - 6 Oct 2014

31 pages

## Keywords

• math.PR
• math.DS
• 37A05, 37A35, 60G10, 60G48

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