Mixing-case process-level limit theory

Establish the mixing-case extension of the process-level Proposition P2′ for the segment quantum mutual-information statistic under bounded strongly mixing sequences.

Background

Proposition P2′ establishes a process-level Gaussian limit away from independence for i.i.d. segments with oracle marginal transforms. The paper remarks that the corresponding strongly mixing result requires an appropriate invariance principle and is not proved, leaving the extension unresolved.

References

The main open problems are theoretical: sharper localisation rates; the mixing-case extension of the process-level Proposition P2′ of Appendix A.6, which needs an invariance principle for bounded strongly mixing sequences [41]; a limit theory for the studentised scan, whose null law stabilises across sample sizes in Supplementary Section B.3; consistency of Holevo-PELT in the number of segments, whose expected behaviour Supplementary Section B.3 also reports; and a detection-delay lower bound via the quantum Fisher information, for which Section 3.4 is the first step.

— Quantum mutual information statistics for detecting dependence-structure change points in time series  (2609.02787 - Kang et al., 2 Sep 2026) in Section 7, Discussion; Appendix A.6

We therefore state P2 under (A0), and we do not claim that 𝜎2 above is the asymptotic variance of the rank-based statistic; deriving the corrected variance is left open.

— Quantum mutual information statistics for detecting dependence-structure change points in time series  (2609.02787 - Kang et al., 2 Sep 2026) in Appendix A.6, Proposition P2 and its rank-based discussion