Extend the framework beyond stationarity and discrete alphabets

Extend the theoretical framework and results to non-stationary stochastic processes and to general (potentially continuous) alphabets by removing the current assumptions that the time series Xt is stationary and takes values on a discrete alphabet, and establish the corresponding definitions and theorems for these broader settings.

Background

In the Appendix, the authors specify that their formal development assumes discrete alphabets and stationarity to simplify results. They note that while extensions are possible, they are not provided in the current work.

Relaxing these assumptions would broaden applicability to many real-world systems that are non-stationary or involve continuous-valued variables, requiring careful reformulation of the core constructs (e.g., causal states, e- and v-machines) and proofs.

References

Extensions for non-stationary processes and/or general alphabets are possible, but will be left for future work.

— Software in the natural world: A computational approach to hierarchical emergence  (2402.09090 - Rosas et al., 2024) in Appendix: 1. Scenario and basic assumptions

Developing multiple change point detection procedures that transcend the MOSUM paradigm and accommodate genuinely nonstationary noise, in which the data-generating mechanism may vary arbitrarily over time subject only to weak dependence and moment assumptions, therefore remains an important and largely open problem.

— Recursive Multiple Change Point Detection of Nonstationary Time Series: Instability Tests, Estimation and Confidence Intervals  (2608.13352 - Cai et al., 13 Aug 2026) in Section 1, Introduction