A gap in the first pass
A new preprint proposes a bootstrap procedure that adds subsampling to frequency-based simulation so it can estimate more of the dependence in functional time series. Its main theoretical result is conditional: when the projection size m, the number of leading eigenfunctions retained, grows at a controlled rate, the procedure consistently estimates the full covariance and relation operators and the distribution of the statistic Ln studied in the paper.
The need for the extra step comes from a limitation in the independent-frequency first stage. That stage asymptotically reproduces the Gaussian component based on Ψ1 and Υ1, but it cannot capture the paper’s Ψ2 and Υ2 terms, which represent the missing fourth-order dependence.
The proposed correction uses a leading-eigenfunction projection and convolved resampling of subsample periodograms. In the projected part of the calculation, that convolved-subsample component consistently estimates the fourth-order terms Ψ2,m and Υ2,m that the first stage leaves out.
There is a qualification when m is kept fixed. The final covariance and relation estimates still contain residual approximation terms alongside the first- and projected fourth-order components. The paper’s full-consistency result therefore depends on letting m increase under the stated growth conditions.
What the theorem requires
The asymptotic theory rests on a formal set of assumptions. It considers a centered, strictly stationary, Hilbert-space-valued process, meaning the model’s statistical behavior is treated as stable over time. The process must have a finite eighth moment, the weighting function W must have bounded variation, and the spectral-density estimator must be uniformly consistent in nuclear norm.
The projection cannot grow arbitrarily quickly. Its allowable rate is tied to the spectral gaps αj. Under polynomial eigenvalue-gap decay, the paper gives the condition m ∼ nβ for β ∈ (0, 1/(2θ + 1)), linking the number of retained components to the series length.
Under those assumptions and the growth rule, the increasing-m procedure consistently estimates the full covariance and relation operators as well as the distribution of Ln. This is a theoretical guarantee within the paper’s stated setting, rather than a claim that every functional time series will meet the conditions.
Synthetic tests show the correction at work
The finite-sample study used synthetic functional time series of length n = 128 in L2([0, 1], R), with observations taken on a grid. It tested three functional autoregressive and moving-average models, labeled FAR(1)/FMA(1), represented with a Fourier basis of D = 25 functions. Model III was the case identified as having a non-vanishing fourth-order spectral density operator.
Exact target values were based on 10,000 repetitions. Bootstrap estimates used a subsampling parameter of b = 32, 300 bootstrap repetitions and 100 evaluation trials.
The two reported criteria for choosing m behaved similarly in 1,000 trials at Q = 0.85. Model I selected m = 2 at a reported frequency of 99.5 for both VRn and DeVRn. Model II most often selected m = 3, at 69.0 for VRn and 70.8 for DeVRn. Model III split between m = 2, at 52.8 and 52.0, and m = 3, at 47.2 and 48.0.
At bandwidth h = 0.14 and grid point τ1, the reported exact-target, bootstrap-mean and bootstrap-standard-deviation figures were 0.296, 0.286 and 0.081 for Model I; 3.279, 3.134 and 0.747 for Model II; and 1.771, 1.826 and 0.310 for Model III. The authors describe the estimates as close to the exact values across the tested bandwidth range.
The simulation comparison also reported visible improvement when convolved subsampling was used to complement the non-corrected bootstrap for the missing fourth-order terms. The quantified result remains a selected finite-sample check at the reported bandwidth and grid point, not a universal benchmark for every tuning choice.
A promising result with a narrow test
The document is labeled arXiv:2608.25765v1 and dated 26 Aug 2026. Its numerical evidence comes from series of length 128 and the three synthetic models described in the study, while the broader validity claim depends on the paper’s regularity and projection-growth assumptions.
The authors report partial funding from the Cyprus Academy of Sciences, Letters, and Arts and partial funding from the Austrian Science Foundation, or FWF.
Paper data and sources
Original title: Frequency Domain Bootstrap for Functional Time Series
Authors: Daniel Rademacher, Jens-Peter Kreiss, Efstathios Paparoditis
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text