Mixing, Ergodic, and Nonergodic Processes With Rapidly Growing Information Between Blocks

We construct mixing processes over an infinite alphabet and ergodic processes over a finite alphabet for which Shannon mutual information between adjacent blocks of length
$n$
grows as
$n^{\beta}$
, where
$\beta\in(0,1)$
. The processes are a modification of nonergodic Santa Fe processes, which were introduced in the context of natural language modeling. The rates of mutual information for the latter processes are alike and also established in this paper. As an auxiliary result, it is shown that infinite direct products of mixing processes are also mixing.