On some bidimensional denumerable chains of infinite order (Q1092539)

We study homogeneous chains of infinite order \((\xi_ t)_{t\in {\mathbb{Z}}}\) with the set of states taken to be \(X=({\mathbb{N}}\setminus \{0,1\})\times \{-1,1\}\). Our approach is to interpret the half-infinite sequence \(...,\xi_{-m}...\xi_{-1},\xi_ 0\), where \(\xi_ t=(i_ t,\epsilon_ t)\in X\), \(t\in {\mathbb{Z}}\), as the continued fraction to the nearer integer expansion (read inversely) of a \(y\in [-,]\). Thus, we are led to study certain Y-valued Markov chains, where \(Y=[-,]\) and then by making use of their properties we establish the existence of denumerable chains of infinite order under conditions different from those given in Theorem 2.3.8 of \textit{M. Iosifescu} and \textit{R. Theodorescu}, Random processes and learning (1969; Zbl 0194.511). A (weak) variant of mixing is proved as well.