  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.
