Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Random-step Markov processes   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.

30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th