We introduce a simple analysis of the structural complexity of
infinite-memory processes built from random samples of stationary, ergodic
finite-memory component processes. Such processes are familiar from the well
known multi-arm Bandit problem. We contrast our analysis with
computation-theoretic and statistical inference approaches to understanding
their complexity. The result is an alternative view of the relationship between
predictability, complexity, and learning that highlights the distinct ways in
which informational and correlational divergences arise in complex ergodic and
nonergodic processes. We draw out consequences for the resource divergences
that delineate the structural hierarchy of ergodic processes and for processes
that are themselves hierarchical.Comment: 8 pages, 1 figure; http://csc.ucdavis.edu/~cmg/compmech/pubs/soi.pd