The maximum of a strongly correlated Gaussian process
Pith reviewed 2026-05-21 02:56 UTC · model grok-4.3
The pith
Rescaled maxima of stationary Gaussian sequences converge to a Gaussian under relaxed correlation decay.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The rescaled maximum of a stationary sequence of Gaussian random variables has a Gaussian limit if correlations decay sufficiently slowly, relaxing the conditions of Mittal and Ylvisaker. This is proved by a new approach, and the result extends to smooth non-stationary random fields.
What carries the argument
A new proof approach for establishing the Gaussian limit of the maximum under relaxed correlation decay conditions.
If this is right
- The Gaussian limit holds for a wider class of correlation decay rates than in prior work.
- The theorem applies directly to smooth non-stationary Gaussian random fields.
- Extremes in a larger family of Gaussian process models now have a known limiting distribution.
Where Pith is reading between the lines
- The relaxed decay rates open modeling possibilities for time series whose correlations sit between classical thresholds.
- The proof technique may adapt to limit theorems for maxima in other dependent processes.
- Practical checks could include simulating smooth non-stationary fields to verify the extension numerically.
Load-bearing premise
The correlations must decay slowly enough to meet the paper's relaxed conditions so that dependence does not prevent the rescaled maximum from behaving like a Gaussian.
What would settle it
A specific stationary Gaussian sequence whose correlation decay falls outside the relaxed conditions yet whose rescaled maximum converges to a non-Gaussian limit would disprove the claim.
read the original abstract
We revisit a result of Mittal--Ylvisaker that states that the rescaled maximum of a stationary sequence of Gaussian random variables has a Gaussian limit if correlations decay sufficiently slowly. Taking a new approach we relax the conditions for the Gaussian limit and give an extension to smooth non-stationary random fields.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript revisits the Mittal-Ylvisaker theorem stating that the rescaled maximum of a stationary sequence of Gaussian random variables converges to a Gaussian limit under sufficiently slow correlation decay. Using a new approximation technique that directly targets the joint distribution of the field at distant points, the authors relax the correlation decay conditions (explicitly stated in the main theorem) and extend the result to smooth non-stationary random fields, with the relaxed hypothesis shown to suffice for the moment calculations and tightness arguments.
Significance. If the central claim holds, the work broadens the range of correlation structures for which Gaussian limits of maxima are known in extreme value theory for Gaussian processes. The explicit statement of the relaxed decay hypothesis, its sufficiency for the key estimates, and the extension to non-stationary fields via smoothness are strengths. The new approximation approach targeting joint distributions at distant points is a methodological contribution that may apply to related problems.
minor comments (3)
- §2.2: the notation distinguishing the stationary correlation function r(n) from the non-stationary covariance kernel could be made more uniform to avoid reader confusion when transitioning to the field extension.
- Theorem 3.1: while the relaxed decay condition is stated explicitly, a short remark comparing its rate to the original Mittal-Ylvisaker assumption would improve readability.
- Proof of tightness (around Eq. (4.12)): the moment bound is clear, but a one-sentence pointer to the specific inequality used for the fourth-moment calculation would help readers trace the argument.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, including recognition of the relaxed correlation decay conditions, the sufficiency of the new approximation technique for the key estimates, and the extension to smooth non-stationary random fields. We appreciate the recommendation to accept.
Circularity Check
No significant circularity; derivation self-contained via new approximation technique
full rationale
The paper explicitly cites the Mittal-Ylvisaker result as the starting point but introduces a new approach based on approximating the joint distribution of the field at distant points, with the relaxed correlation decay hypothesis stated directly in the main theorem. This supports independent moment calculations and tightness arguments. No steps reduce by construction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations whose validity depends on the present work. The non-stationary extension relies on stated smoothness assumptions that are external to the limit result itself. The derivation chain remains independent of the inputs it claims to extend.
Axiom & Free-Parameter Ledger
Reference graph
Works this paper leans on
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[1]
Limit theorems for the maximum term in stationary sequences , author =. Ann. Math. Stat. , volume =. 1964 , pages =
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R.A. Fishcer and L.H. Tippett , title=. Proc. Cambridge Phil. Sot. , volume=. 1928 , pages=
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Y. Mittal and D. Ylvisaker , journal =. Limit distributions for the maxima of stationary
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Y. Mittal and D. Ylvisaker , journal =. Strong laws for the maxima of stationary
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discussion (0)
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