## The Annals of Probability

- Ann. Probab.
- Volume 21, Number 4 (1993), 2157-2168.

### On 1-Dependent Processes and $k$-Block Factors

Robert M. Burton, Marc Goulet, and Ronald Meester

#### Abstract

A stationary process $\{X_n\}_{n \in \mathbb{Z}}$ is said to be $k$-dependent if $\{X_n\}_{n < 0}$ is independent of $\{X_n\}_{n > k-1}$. It is said to be a $k$-block factor of a process $\{Y_n\}$ if it can be represented as $X_n = f(Y_n,\ldots, Y_{n+k-1}),$ where $f$ is a measurable function of $k$ variables. Any $(k + 1)$-block factor of an i.i.d. process is $k$-dependent. We answer an old question by showing that there exists a one-dependent process which is not a $k$-block factor of any i.i.d. process for any $k$. Our method also leads to generalizations of this result and to a simple construction of an eight-state one-dependent Markov chain which is not a two-block factor of an i.i.d. process.

#### Article information

**Source**

Ann. Probab., Volume 21, Number 4 (1993), 2157-2168.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989014

**Digital Object Identifier**

doi:10.1214/aop/1176989014

**Mathematical Reviews number (MathSciNet)**

MR1245304

**Zentralblatt MATH identifier**

0788.60049

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G10: Stationary processes

Secondary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 28D05: Measure-preserving transformations

**Keywords**

Stationary processes one-dependence block factors Markov chains

#### Citation

Burton, Robert M.; Goulet, Marc; Meester, Ronald. On 1-Dependent Processes and $k$-Block Factors. Ann. Probab. 21 (1993), no. 4, 2157--2168. doi:10.1214/aop/1176989014. https://projecteuclid.org/euclid.aop/1176989014