Chapter 10: Processes of Perception and Analysis

Section 9: Statistical Analysis

[Models involving] non-local processes

It follows from the fact that any path in a finite network must always eventually return to a node where it has been before that any Markov process must be fundamentally local, in the sense that the probabilities it implies for what happens at a given point in a sequence must be independent of those for points sufficiently far away. But probabilistic models based on other underlying systems can yield sequences with long-range correlations. As an example, probabilistic neighbor-independent substitution systems can yield sequences with hierarchical structures that have approximate nesting. And since the mid-1990s such systems (usually characterized as random trees or random context-free languages) have sometimes been used in analyzing data that is expected to have grammatical structure of some kind.

From Stephen Wolfram: A New Kind of Science [citation]