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We introduce a simple analysis of the structural complexity ofinfinite-memory processes built from random samples of stationary, ergodicfinite-memory component processes. Such processes are familiar from the wellknown multi-arm Bandit problem. We contrast our analysis withcomputation-theoretic and statistical inference approaches to understandingtheir complexity. The result is an alternative view of the relationship betweenpredictability, complexity, and learning that highlights the distinct ways inwhich informational and correlational divergences arise in complex ergodic andnonergodic processes. We draw out consequences for the resource divergencesthat delineate the structural hierarchy of ergodic processes and for processesthat are themselves hierarchical.

Signatures of Infinity: Nonergodicity and Resource Scaling in Prediction, Complexity, and Learning