Statistics and modelling of order patterns in univariate time series

Christoph Bandt

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Proposition 6 defines the same multiplicative gluing rule with a 1/2 split when the end letters are adjacent in value, and proves it extends P_m and yields stationarity; it then iterates and constructs a stationary random order, using an interval–martingale argument for existence. The candidate solution reproduces this construction, supplies a clean combinatorial counting lemma that justifies the 1/2 split and double-counting issue, verifies normalization and both first-/last-window marginals, and then uses the Daniell–Kolmogorov extension for existence. Aside from explicitly handling the zero-denominator case and using DK instead of the paper’s martingale device, the arguments align closely with equation (28) and Proposition 5 in the paper and are correct .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The extension rule and stationarity checks are correct and practically important. The paper presents a clear formula and a concise proof, and a nice existence argument via martingale convergence on an interval coding of orders. Two small clarifications (zero-denominator handling and a brief combinatorial justification of the 1/2 split) would make the result fully airtight and easier to apply. With these tweaks, the contribution provides a solid foundation for extending empirical short-pattern distributions to full stationary ordinal models.