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arxiv: 2607.01931 · v1 · pith:ZKMPDOMYnew · submitted 2026-07-02 · 🧮 math.ST · stat.ME· stat.TH

Transferring supremum-norm rates and weak convergence of covariance kernel estimators to functional principal components

Hajo Holzmann , Kevin Wilk This is my paper

Pith reviewed 2026-07-03 04:13 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords functional principal componentssupremum normperturbation theorycovariance kernelasymptotic normalityminimax ratesfunctional data analysissparse design
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The pith

L2-perturbation theory transfers supremum-norm rates and weak convergence from covariance kernel estimators to functional principal components.

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

The paper shows that L2-perturbation theory moves rates of convergence in the supremum norm and weak convergence in the space of continuous functions from covariance kernel estimators directly to the associated functional principal components. In a discrete observational model with errors under fixed synchronous design, this produces optimal sup-norm rates for the FPCs, including minimax lower bounds, together with asymptotic normality. The same transfer captures the sparse-to-dense transition previously seen for means and kernels, and it extends to cross-covariance kernels, long-run covariance kernels, and kernels of derivative processes. Eigenvalue estimation under sparse designs remains discretization-dominated. A reader would care because the result supplies a single mechanism that converts existing kernel results into corresponding FPC results without new case-by-case arguments.

Core claim

L2-perturbation theory can be used to transfer rates of convergence in the supremum norm as well as weak convergence in the space of continuous functions from covariance kernel estimators to the associated functional principal components. As an application, optimal rates of convergence in sup-norm, including minimax lower bounds, as well as asymptotic normality are obtained for estimating the FPCs in a discrete observational model with errors under fixed, synchronous design. The sparse to dense transition applies to the FPCs, and eigenvalue estimation exhibits a discretization-dominated regime under sparse designs. The results further apply to estimators of cross-covariance and long-run cova

What carries the argument

L2-perturbation theory that lifts sup-norm rates and C[0,1] weak convergence of a kernel estimator to the eigenfunctions and eigenvalues of the associated integral operator.

If this is right

  • Optimal sup-norm rates, including minimax lower bounds, hold for the functional principal components under the discrete observational model with errors.
  • Asymptotic normality holds for the functional principal component estimators in the same setting.
  • The sparse-to-dense transition observed for mean and covariance estimators also holds for the functional principal components.
  • Eigenvalue estimation remains in a discretization-dominated regime under sparse designs.
  • The transfer applies equally to cross-covariance kernels, long-run covariance kernels, and covariance kernels of derivative processes.

Where Pith is reading between the lines

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

  • The same perturbation argument could be applied to obtain sup-norm results for other linear functionals of the covariance operator once kernel rates are known.
  • Numerical implementation via the Nyström method, already tested in the paper, offers a direct way to check whether the transferred rates appear in finite samples under varying sampling densities.
  • Because the method requires no extra eigenvalue-gap or smoothness conditions beyond those already present in the kernel estimators, it may simplify proofs for related objects such as canonical correlation functions or functional regression coefficients.
  • The discretization-dominated regime for eigenvalues under sparse designs suggests that further work on optimal sampling designs could focus on eigenvalue rather than eigenfunction recovery.

Load-bearing premise

The covariance kernel estimators are already assumed to satisfy the required supremum-norm rates and weak convergence in the space of continuous functions.

What would settle it

A concrete counter-example in which a covariance kernel estimator meets the stated sup-norm rates and C[0,1] convergence yet the associated functional principal component estimator fails to meet the transferred rates.

Figures

Figures reproduced from arXiv: 2607.01931 by Hajo Holzmann, Kevin Wilk.

Figure 1
Figure 1. Figure 1: Estimation of principal component basis functions and corresponding eigenvalues: [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean of supremum norm error of N = 1,000 iterations with estimator Γˆ 500 of degree one for estimating the principal component basis functions (41) and covariance kernel (40) of the OU-process. (a) Mean of ∥ϕˆk;n,201(· ; h ∗ ) − cˆk;201 ϕk∥∞. (b) Mean of |(λˆk;n,201(h ∗ ) − λk)/λk| [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of increasing p, where for each combination of n and p the optimal h ∗ is determined similar to Figures 2(a) and 2(b). 4.3 Daily temperature curves in Berlin As an empirical illustration we revisit a dataset analyzed in Wilk and Holzmann (2026), and consider time series of daily air temperatures for each month from 2000 to 2025 at a weather station in Berlin, Germany3 . Thus we have 26 segments for … view at source ↗
Figure 4
Figure 4. Figure 4: Estimation of FPCs with 95% uniform confidence bands and corresponding eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Estimates of the FPCs with 95% uniform confidence bands and corresponding eigen [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Covariance estimation (blue–yellow colored area) and covariance kernel (red colored [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Analogous to Figure [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Analogous to Figure [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
read the original abstract

We show that $L_2$-perturbation theory can be used to transfer rates of convergence in the supremum norm as well as weak convergence in the space of continuous functions from covariance kernel estimators to the associated functional principle components (FPCs). As an application we obtain optimal rates of convergence in sup-norm, including minimax-lower bounds, as well as asymptotic normality for estimating the FPCs in a discrete observational model with errors under fixed, synchronous design. The sparse to dense transition which has previously been observed for mean function and covariance kernel estimators also applies to the FPCs. Surprisingly, eigenvalue estimation exhibits a discretization-dominated regime under sparse designs, too. Our results further apply to estimators of cross-covariance and long-run covariance kernels, as well as to covariance kernels of derivative processes. We also present results of numerical experiments in which we use the Nystr\"om method to compute FPCs and eigenvalues, and give an empirical illustration to series of daily temperature curves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper shows that L2-perturbation theory transfers supremum-norm convergence rates and weak convergence in C[0,1] from covariance kernel estimators to the associated functional principal components. As an application in a discrete observational model with errors under fixed synchronous design, optimal sup-norm rates (including minimax lower bounds) and asymptotic normality are obtained for FPC estimators. The sparse-to-dense transition extends to FPCs, eigenvalue estimation is discretization-dominated under sparse designs, and the results apply to cross-covariance, long-run covariance, and derivative-process kernels. Numerical experiments use the Nyström method, with an empirical illustration on daily temperature curves.

Significance. If the transfer mechanism holds, the work supplies a general, reusable tool for deriving sup-norm rates and weak convergence for FPCs from existing kernel-estimator results, which is a useful contribution to functional data analysis. The discrete-design application yields new optimal rates, minimax bounds, and normality results for FPCs together with the sparse-to-dense and discretization observations; the extensions to other kernels broaden applicability. Credit is due for the clean perturbation-based transfer approach and the concrete numerical/empirical validation.

minor comments (3)
  1. [§2.2] §2.2: the statement of the L2-perturbation result should explicitly record the eigenvalue-gap and eigenfunction-smoothness conditions needed for the transfer to preserve the sup-norm rate and the C[0,1] weak-convergence topology.
  2. [§4.3] §4.3 (application): the minimax lower-bound argument for the FPCs is obtained by transfer; a short remark confirming that the perturbation constants do not degrade the lower-bound constant would strengthen the claim.
  3. [Numerical experiments] Figure 2 caption: the simulation design densities and the number of Monte Carlo replications should be stated explicitly so that the reported sup-norm errors can be reproduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the clear summary of our contributions, and the recommendation of minor revision. We are pleased that the perturbation approach is viewed as a reusable tool and that the discrete-design results, sparse-to-dense transition, and extensions are recognized as useful.

Circularity Check

0 steps flagged

No significant circularity; external perturbation theory applied to assumed inputs

full rationale

The paper's central derivation invokes L2-perturbation theory to transfer sup-norm rates and weak convergence from covariance kernel estimators (treated as given inputs satisfying the required properties) to the associated FPCs. This structure is self-contained against external benchmarks because the kernel rates are not re-derived from the FPC conclusions, no parameters are fitted to subsets and then renamed as predictions, and no load-bearing uniqueness theorem or ansatz is smuggled via self-citation. The application to the discrete observational model and minimax bounds follows directly from the transfer without reducing the target result to its own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or ad-hoc axioms beyond the invocation of L2-perturbation theory; the claim rests on the existence of suitable rates for the input kernel estimators and standard functional-analytic perturbation results.

axioms (1)
  • domain assumption L2-perturbation theory applies to transfer sup-norm rates and weak convergence from covariance kernels to their eigen-elements
    Invoked in the first sentence of the abstract as the mechanism for the transfer.

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