A note on the Bramson–Kalikow process

Abstract

We consider discrete-time stationary processes with long-range dependencies, $X_{n}\in\{\pm1\}$, ${n\in\mathbb{Z}}$, specified by a regular attractive $g$-function, similar to those considered by Bramson and Kalikow [Israel J. Math. 84 (1993) 153–160]. We give an explicit set of conditions that imply the existence of at least two distinct processes specified by the same $g$-function, and consider a few examples that emphasize the role played by the smoothness of the majority rule at the origin.