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

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.