Informational and Causal Architecture of Discrete-Time Renewal Processes

TL;DR

This paper develops a theoretical framework for understanding the informational and causal structure of discrete-time renewal processes, including their minimal sufficient statistics, memory requirements, and entropy decomposition.

Contribution

It introduces a new subclass of renewal processes with finite causal states despite unbounded interevent times and provides formulas for analyzing processes with infinite complexity.

Findings

Derived formulas for causal states and statistical complexity.

Identified a subclass with finite causal states despite unbounded interevent durations.

Analyzed the Simple Nonunifilar Source with infinite-state epsilon-machine.

Abstract

Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate the historical memory capacity required to store those states (statistical complexity), delineate what information is predictable (excess entropy), and decompose the entropy of a single measurement into that shared with the past, future, or both. The causal state equivalence relation defines a new subclass of renewal processes with a finite number of causal states despite having an unbounded interevent count distribution. We use these formulae to analyze the output of the parametrized Simple Nonunifilar Source, generated by a simple two-state hidden Markov model, but with an infinite-state epsilon-machine presentation. All in all, the results lay the groundwork for analyzing processes with infinite statistical complexity and infinite excess entropy.