A Functional Central Limit Theorem for Localized Partial Sums of Non-Stationary Time Series

Florian Heinrichs

REVIEW 3 major objections 4 minor 20 references

Kernel-weighted partial sums of non-stationary time series converge weakly to a Gaussian random distribution in D'(0,1).

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2026-08-01 17:16 UTC pith:6ZN2QIYC

load-bearing objection The D'-valued formulation is a nice idea, but the central theorem is normalized inconsistently: under the displayed definition the limit is degenerate, and the proof and applications quietly switch to a different scaling. the 3 major comments →

arxiv 2607.17697 v1 pith:6ZN2QIYC submitted 2026-07-20 math.ST stat.TH

A Functional Central Limit Theorem for Localized Partial Sums of Non-Stationary Time Series

Florian Heinrichs This is my paper

classification math.ST stat.TH MSC 60F1762M1060G6562G20

keywords Physical Dependence MeasurePiecewise Local StationarityFunctional Central Limit TheoremKernel-Weighted Partial SumsChange DetectionDistribution-Valued Weak ConvergenceIsonormal Gaussian ProcessNonparametric Regression

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a functional central limit theorem for localized kernel-weighted partial sums of piecewise locally stationary time series. Rather than pointwise convergence at each location, the entire process is shown to converge weakly in the space of distributions D'(0,1) to a centered Gaussian random distribution. The limiting covariance is the integral of the time-varying long-run variance against the test functions, and the result extends to isonormal Gaussian process limits on L² and to uniform convergence over totally bounded function classes. The theory is then used to construct level-α tests for a constant mean against linear, polynomial, and arbitrary alternatives in nonparametric regression with locally stationary errors, supported by simulations and real temperature and EEG data.

Core claim

The paper's central claim is a distribution-valued functional central limit theorem: for a piecewise locally stationary time series with geometrically decaying physical dependence measure and a compactly supported Lipschitz kernel K, the localized partial sum process G_n(t) = (n h_n)^{-1/2} ∑_{i=1}^n (X_{i,n} − E X_{i,n}) K((i/n − t)/h_n) converges weakly in D′(0,1) to a centered Gaussian random distribution G whose finite-dimensional covariance is Cov(⟨φ_i,G⟩,⟨φ_j,G⟩) = ∫_0^1 σ²(x) φ_i(x) φ_j(x) dx, provided n^{1/4} h_n / log² n → ∞. At continuity points of σ² this matches the pointwise local variance formula, but the distribution-valued statement goes beyond pointwise asymptotics to cover

What carries the argument

The load-bearing mechanism is a Gaussian approximation for piecewise locally stationary processes with geometrically decaying physical dependence measure: it replaces the original non-linear filter-driven process by a sum of independent normals with time-varying variances, with an error bound of order o_P(n^{1/4} log² n). After this replacement, the variance of any linear functional ⟨φ, G_n⟩ reduces to an explicit Riemann-sum integral converging to ∫ σ² φ². Distribution-valued weak convergence is formalized through smooth test functions on (0,1), finite-dimensional marginals, and a tightness criterion for random distributions, while an L² bound on ∥G_n∥_{L²} supplies the uniform equicontinui

Load-bearing premise

The proof assumes a normalization of the localized sums that keeps the variance of every fixed projection of order one; the definition as written in the paper does not provide that normalization, so the theorem stands or falls on this normalization being corrected.

What would settle it

Compute the second moment of ⟨φ, G_n⟩ using the normalization G_n(t) = (n h_n)^{-1/2} ∑ (X_{i,n} − E X_{i,n}) K((i/n − t)/h_n) for a fixed smooth test function φ. The variance tends to 0, not to ∫ σ² φ² dx, which would contradict the non-degenerate covariance claimed in Theorem 1; the proof's variance computation uses a different normalization, so a referee can check which convention the theorem is actually proved for.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Projections of the localized process onto any L² function converge to Gaussian variables with variance ∫ σ² f², so level-α tests for linear and polynomial trends can be built with explicit normal or chi-square critical values, without bootstrap or self-normalization.
The random-projection test for a constant mean against an arbitrary alternative is consistent: any non-constant mean is detected with probability one, conditional on the random test function.
Because the limit lives in the space of distributions, the theory covers statistics that integrate the entire localized process over the interval, not just evaluations at a single point.
When the set of admissible test directions is totally bounded in L²([0,1]), the empirical process indexed by that set converges in ℓ∞, permitting data-driven or optimized choices of test functions.
The same Gaussian limit underpins bias-corrected local linear estimation, so inference on the mean function inherits the volatility-adaptive variance structure from σ²(·).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

A natural next step, not addressed in the paper, is to adapt the distribution-valued limit to local polynomial estimators of order higher than one; the same Gaussian approximation should yield an analogous covariance structure once the bias is properly removed.
The totally bounded family condition is mild enough that the ℓ∞ result may carry over to Donsker-type function classes under the same physical dependence assumptions, which would let the test direction be chosen adaptively from a rich dictionary without paying for multiple comparisons.
The random-projection construction suggests an extension where several independent test directions are averaged or combined to reduce finite-sample conservativeness; one could check whether sub-Gaussian coefficients such as Rademacher variables preserve the asymptotic level while improving power.
If the bandwidth condition is tuned, the theory might extend to kernels without compact support, provided the kernel tails decay fast enough to keep the variance integral finite.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The D'-valued formulation is a nice idea, but the central theorem is normalized inconsistently: under the displayed definition the limit is degenerate, and the proof and applications quietly switch to a different scaling.

read the letter

Quick take: the paper has a real idea — viewing localized partial sums as random distributions and proving a process-level Gaussian limit — but the central theorem is false under the normalization that is displayed. The proof and the applications quietly switch to a different scaling. This is a load-bearing flaw, not a typo.

What is genuinely new: the D'(0,1) formulation, the L2-indexed extension, and the ℓ∞(G) tightness argument are not in the cited literature. The paper is clearly written, the simulations are extensive, code is available, and the authors are transparent about finite-sample distortions. If the normalization is fixed, the proof route — Zhou's Gaussian approximation plus standard tightness — is standard and likely goes through.

The problem: §3 defines G_n(t) = (n h_n)^(-1/2) Σ ... . For any test function φ, Var(⟨φ,G_n⟩) ≈ h_n ∫ σ² φ², so ⟨φ,G_n⟩ → 0 and the D'-limit is degenerate. The claimed covariance ∫ σ² φψ cannot hold under this definition. In §6.1, Eq. (13) computes the variance with 1/(n h_n²), the square of 1/(√n h_n); in Section 4 the localized error process is also written with 1/(√n h_n). So the theorems are plausibly true for the proof's normalizer but not for the stated one. The statement of Theorem 1(2), Theorem 2, Theorem 3, and the test statistics all inherit the ambiguity. This is not a cosmetic issue: the covariance structure and the rate condition n^{1/4} h_n / log² n → ∞ depend on it.

Minor points: Theorem 2's proof leans on Proposition 13 of Heinrichs (2025) for the covariance bound, which is not reproduced; that is acceptable but should be checked. The simulations show the new tests are occasionally anti-conservative, which the authors report honestly, but it weakens the empirical case.

Bottom line: the distribution-valued formulation is worth pursuing and may be salvageable, but in its current form the main theorem is false as stated. I would not cite it yet. I would not desk-reject it outright, though: the flaw is identifiable and fixable, and a referee could do useful work. But the reader's reject verdict is right for the manuscript as it stands.

Referee Report

3 major / 4 minor

Summary. The paper develops a functional central limit theorem for kernel-localized partial sums of piecewise locally stationary time series with geometrically decaying physical dependence. The claimed limit is a centered Gaussian random distribution on D'(0,1) with covariance ∫ σ² φψ, extended to L²([0,1])-indexed projections and to ℓ∞(G) for totally bounded G ⊂ L²([0,1]). The results are then applied to tests for a constant mean in nonparametric regression with locally stationary errors, supported by simulations and data examples. The proof strategy is based on the Gaussian approximation of Zhou (2013) and Wu–Zhou (2011), followed by variance calculations for the localized sums.

Significance. If established, the result would be a useful process-level extension of pointwise CLTs for localized sums under local stationarity, and it would provide a principled distribution-valued framework for projection-based change-point tests. The paper is clearly written, is well connected to the literature, and ships reproducible Python code and empirical results, which are strengths. However, the central theorem as stated is not valid under the displayed normalization, and the proofs of the L²-extension and the ℓ∞(G)-tightness rely on a false second-moment bound. These are load-bearing problems that must be resolved before the claims can be accepted.

major comments (3)

[§3 and §6.1 (Eq. 13)] The normalization of G_n is inconsistent. As displayed, G_n(t) = (n h_n)^{-1/2} Σ (X_{i,n}-EX_{i,n}) K_{h_n}(i/n - t). For i.i.d. errors, Var⟨φ,G_n⟩ = (n h_n)^{-1} Σ σ²(i/n)(∫φK_{h_n})² = O(h_n) → 0, so the limit in Theorem 1(2) would be degenerate; similarly √h_n G_n(t) has variance O(h_n), contradicting Theorem 1(1). The proof in Eq. (13) instead uses variance (n h_n²)^{-1} Σ..., which corresponds to the normalizer (√n h_n)^{-1}, and Section 4's G_n^ε is written with that same normalizer. Please choose one normalizer, state it consistently, and revise Theorem 1(1) and its proof accordingly; as written, the proof after (12) does not apply the √h_n factor to the Gaussian approximant.
[§6.2 (Eq. 15)] The bound E||G_n||²_{L²} ≤ C_K is false under every normalization that makes the covariance in Theorem 1 nondegenerate. For instance, under the normalizer (√n h_n)^{-1} used in the proof, the diagonal i=j term alone is (n h_n²)^{-1} Σ σ²(i/n) ∫K_{h_n}(·-i/n)² dt = (n h_n²)^{-1} · n h_n ∫K² = O(1/h_n). Thus Eq. (15) does not hold. This bound is the basis for the Cauchy–Schwarz approximation step in the proof of Theorem 2, so the claimed extension from C_c∞(0,1) to L²([0,1]) is not established by the given argument. A direct Riemann-sum variance computation for the projections would be needed.
[§6.3 (Theorem 3)] The asymptotic tightness proof of Theorem 3 uses the invalid bound from Eq. (15). Specifically, the final Markov bound P(sup_{||f-g||_{L²}<δ} |H_n(f)-H_n(g)| > ε) ≤ C_K δ²/ε² is derived from E||G_n||² ≤ C_K. If the correct order is E||G_n||² = O(1/h_n), then the right-hand side does not tend to zero and no uniform L²-equicontinuity follows. Consequently Theorem 3, and the ℓ∞(G)-convergence result, are unproved as they stand.

minor comments (4)

[§6.4] The proofs of the results from Section 4 do not include a proof of Proposition 4; only Proposition 5 and Proposition 6 are proved. If Proposition 4 is meant to follow by the same argument, this should be stated.
[§4.3 / Proposition 6] Consistency under the general alternative requires that the random projection Φ has a nonzero coefficient for every basis direction that appears in g. If some a_k = 0, then for g = e_k the variance of ⟨Φ,g⟩ is zero and the test has no power in that direction; the assumption on the sequence (a_k) should be made explicit (e.g., a_k ≠ 0 for all k).
[§3–§4] The notation K_{h_n}(·) = K(·/h_n) is introduced only in Section 4 but is already used in Section 3. Please define it in Section 3.
[§5.1] Table 1 shows substantial size distortion for the proposed tests in several settings (e.g., 28.6% for test (10) at n=1000, σ0, i.i.d. errors). The text mentions this but does not offer an explanation; a comment on the source of the distortion would help readers.

Circularity Check

0 steps flagged

No significant circularity: the derivation rests on external Gaussian approximation results; the normalization mismatch is a correctness issue, not a circular reduction.

full rationale

The central FCLT is not circular. Theorem 1 is proved by invoking the Gaussian approximation of Zhou (2013) and Wu and Zhou (2011), which are external results, and then computing the covariance of the Gaussian proxy directly. No fitted parameter is later relabeled as a prediction, and no uniqueness theorem from the author's prior work is used to force the choice of limit. The only self-citation, Proposition 13 of Heinrichs (2025), supplies a covariance bound used to control E||G_n||² in the L² extension and tightness; it is a parameter-free bound from prior work that does not assert the present theorem, so it counts as independent support under the stated rules. A serious normalization inconsistency does exist: Section 3 displays G_n(t) = (n h_n)^{-1/2} Σ (X_i,n−EX_i,n)K_h(i/n−t), while the proof of Theorem 1 in Section 6.1 writes the variance of ⟨φ,G_n⟩ as Σ_n^{(1)} = (n h_n²)^{-1} Σ σ²(∫φK_h)², which is the scaling appropriate to (√n h_n)^{-1}, not (n h_n)^{-1/2}. If the displayed scaling is taken literally, the variance of ⟨φ,G_n⟩ is of order h_n and the claimed D'-limit is degenerate. This makes Theorem 1(2) false as stated, but it is a scaling/correctness error rather than a circularity: the proof does not assume the conclusion, and correcting the scaling would yield a degenerate limit instead of a derivation that returns the target covariance by construction. The simulation-based application sections do not rename fitted quantities as predictions; they use the (problematic) limit theorem to derive test statistics that are then checked numerically. Overall, no derivation step reduces to its own input by definition or by a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no new entities or fitted parameters. The central claim depends on a chain of external Gaussian-approximation and expansion results; the most fragile items are the unproved transfer in (12) and the self-cited covariance bound used for the L² second-moment estimate.

axioms (5)

domain assumption Assumptions 1 and 2: symmetric Lipschitz kernel, piecewise Lipschitz L2 continuity, finite fourth moments, geometric decay δ_4(h)=O(χ^h), long-run variance bounded away from 0.
These assumptions define the class of kernels and PLS time series for which the theorems are claimed.
domain assumption Proposition 5 of Zhou (2013): Gaussian approximation of PLS partial sums with error o_P(n^{1/4} log² n).
Used at the start of §6.1 to replace centered observations by σ(i/n)V_i; the paper does not reproduce the proof.
domain assumption Equation (12): transfer of the Gaussian approximation to kernel-weighted sums via summation by parts.
The transfer is asserted with a citation to Bücher et al. (2021); uniformity in t and in test functions is not shown in this paper.
domain assumption Proposition 13 of Heinrichs (2025): |Cov(X_i,n, X_j,n)| ≤ Θ|i-j|^{-1}.
Self-cited in the proof of Theorem 2 to bound E||G_n||²_{L²}; no proof or independent verification is provided here.
domain assumption Lemma C.2 of Dette and Wu (2019): expansion ilde μ_n - μ = n^{-1/2} G_n^ε + O(h_n³ + 1/(n h_n)).
This expansion underlies all Section 4 test statistics, but the normalization of G_n^ε in the expansion is not stated explicitly and appears inconsistent with the Section 3 definition.

pith-pipeline@v1.3.0-alltime-deepseek · 23155 in / 28037 out tokens · 280777 ms · 2026-08-01T17:16:23.489452+00:00 · methodology

0 comments

read the original abstract

A localized functional central limit theorem is established for kernel-weighted partial sum processes of piecewise locally stationary time series under geometric decay of the physical dependence measure. The localized process is shown to converge weakly to a centered Gaussian random distribution in $D'(0,1)$, and the limit extends naturally to an isonormal Gaussian process on $L^2([0,1])$. Weak convergence is further derived for processes indexed by totally bounded subsets of $L^2([0,1])$. As an application, the localized limit theory is used to construct tests for constant mean functions against linear, polynomial, and general alternatives in non-parametric regression with locally stationary errors. Simulation results and data examples illustrate the finite sample performance and practical applicability of the proposed methodology.

discussion (0)

Reference graph

Works this paper leans on

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arXiv preprint arXiv:2503.14322 (2025) B¨ ucher, A., Dette, H., Heinrichs, F.: Are deviations in a gradually varying mean relevant? a testing approach based on sup-norm estimators

Afonso, T.V., Heinrichs, F.: Consumer-grade eeg-based eye tracking. arXiv preprint arXiv:2503.14322 (2025) B¨ ucher, A., Dette, H., Heinrichs, F.: Are deviations in a gradually varying mean relevant? a testing approach based on sup-norm estimators. The Annals of Statistics 49(6), 3583–3617 (2021) Chac´ on, G., Rafeiro, H., Vallejo, J.C.: Functional Analys...

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Gruyter, Berlin, Boston (2017)

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Dudley, R.: Sample functions of the gaussian process. The Annals of Probability, 66–103 (1973)

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Journal of time series analysis (2025)

Heinrichs, F., Bastian, P., Dette, H.: Sequential outlier detection in nonstationary time series. Journal of time series analysis (2025)

2025

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Electronic Journal of Statistics15(2), 3762–3797 (2021)

Heinrichs, F., Dette, H.: A distribution free test for changes in the trend function of locally stationary processes. Electronic Journal of Statistics15(2), 3762–3797 (2021)

2021

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Heinrichs, F.: Self-normalization for cusum-based change detection in locally station- ary time series. arXiv preprint arXiv:2509.07112 (2025)

Pith/arXiv arXiv 2025

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arXiv preprint arXiv:2505.02197 (2025)

Palm, N., Nagler, T.: Uniform central limit theorems for non-stationary processes via relative weak convergence. arXiv preprint arXiv:2505.02197 (2025)

Pith/arXiv arXiv 2025

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Bernoulli28(1), 453–480 (2022)

Phandoidaen, N., Richter, S.: Empirical process theory for locally stationary processes. Bernoulli28(1), 453–480 (2022)

2022

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Springer Series in Statistics

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2023

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Proceedings of the National Academy of Sciences102(40), 14150–14154 (2005)

Wu, W.B.: Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences102(40), 14150–14154 (2005)

2005

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Statistica Sinica, 1397–1413 (2011) 22

Wu, W.B., Zhou, Z.: Gaussian approximations for non-stationary multiple time series. Statistica Sinica, 1397–1413 (2011) 22

2011

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Journal of the American Statistical Association108(502), 726–740 (2013)

Zhou, Z.: Heteroscedasticity and autocorrelation robust structural change detection. Journal of the American Statistical Association108(502), 726–740 (2013)

2013

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Zhou, Z., Wu, W.B.: Local linear quantile estimation for nonstationary time series. The Annals of Statistics, 2696–2729 (2009) Appendix A Additional Empirical Results This section contains additional results for the experiments from Section 5.1 under the modelX i,n =µ(i/n) +σ(i/n)ε i. In addition to the choicesσ 0 andσ 1, we considered σ2(x) = 1 4 + x 2 a...

2009