{"id":"f80abecc-062a-460f-a9d7-07bd7b0737a6","arxiv_id":"2512.10467","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A bootstrap-based method with Gaussian approximation guarantees asymptotically uniform false discovery rate control for time-varying correlation networks.","lead":"This paper builds a statistical framework that controls the false discovery rate when testing time-varying correlations in high-dimensional, non-stationary time series with jumps in the mean. It combines robust difference-based estimators, multiplier bootstrap, and Benjamini–Yekutieli/Benjamini–Hochberg procedures to produce correlation networks with uniform-in-time error control.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's uniform FDR control rests on Lemma 3, a hyperrectangular Gaussian approximation whose proof is only sketched via 'minor modifications' to Wu et al. (2024); the claimed dimension-free O((nb)^{-1/8}(log n)^4) rate is not established.","rationale":"The reader's CONDITIONAL verdict was based primarily on Lemma 3 being adapted from Wu et al. (2024) with minor modifications and Lemma 1's proof being omitted. My stress-test agrees: these are the weakest points in the chain leading to Theorem 4. I do not see an internal contradiction in the FDR argument itself; the B-Y proof, the summation-by-parts, and the handling of the discretization error 1/B all appear coherent. The simulations and real-data analyses provide supporting evidence, but they use modest n and p and therefore cannot verify the asymptotic dimension-uniform rate claimed in Lemma 3. The abstract's AuFDR definition mismatch is real but secondary: it concerns what is being advertised rather than whether the body's stated uniform-FDR theorem is proved. Thus the principal unresolved risk is the unsubstantiated Gaussian-approximation lemma, which is exactly the kind of technical premise that should be fully proved or explicitly imported with all modifications verified before the central claim is accepted. The appropriate verdict remains CONDITIONAL, requiring the authors to supply a complete proof of Lemma 3 (and Lemma 1) or to state Theorem 4 as conditional on those lemmas.","tokens_in":30464,"tokens_out":19186,"duration_ms":192563,"concrete_test":"Independently re-derive Lemma 3 from Propositions 6–7 of Wu et al. (2024), tracking the lag-h dependence (\\delta_q=O(\\chi^{r-h})) and sub-Weibull tails (E\\exp(t|H|^\\kappa)<\\infty) through inequality (28), and confirm that the final hyperrectangle approximation error is O((nb)^{-1/8}(\\log n)^4) uniformly in x\\in\\mathbb{R}^{p^*} with no polynomial-in-p factor. If the bound gains a factor p^c (c>0), then the condition p^2(\\theta_n^{(1)}+\\theta_n^{(2)}) \\to 0 in Theorems 4–5 fails for p\\asymp n^\\iota, and the uniform FDR guarantee would not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 4) is a sup_t FDR bound, and its proof reduces, via Theorem 3, to simultaneous validity of bootstrap P-values. The linchpin is Lemma 3 (Appendix E.3), a hyperrectangular Gaussian approximation for the time×pairs vector \\bar\\Xi^G_j. Lemma 3 is not proved in this paper; it is asserted to follow from Propositions 6–7 of Wu et al. (2024) with 'minor modifications' after a 'very careful investigation' (p.31). Those modifications are not trivial: the innovations are lag-h differences, so Lemma 2(iii) gives \\delta_q(H_{i,l,\\Xi},r)=O(\\chi^{r-h}), not O(\\chi^r), and the tail condition is sub-Weibull rather than exponential. The displayed error bound (28) acquires terms (\\phi^q n p^* \\chi^{(M-h)q})^{1/(1+q)} and exp(-t M_x^\\kappa), and the final sup over hyperrectangles is claimed to be O((nb)^{-1/8}(\\log n)^4) with no explicit p-dependence. Theorem 1 uses Lemma 3 at face value and Theorem 3's P-value validity inherits its rate; an uncounted polynomial-in-p factor would destroy the proof's condition p^2(\\theta_n^{(1)}+\\theta_n^{(2)})\\to 0 in Theorems 4–5. Lemma 1, also needed for Theorem 1, has its proof omitted ('For simplicity we omit the proof,' Appendix E.1). These are not peripheral details; they are the load-bearing bridge from the bootstrap distribution to P-value uniformity.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a bootstrap-based procedure for testing time-varying pairwise correlations in high-dimensional locally stationary time series with possible jumps in the means, and proves uniform-in-time asymptotic false discovery rate control for the resulting time-varying correlation networks. The main theoretical engine is a hyperrectangular Gaussian approximation for the product sequence of two locally stationary processes (Lemma 3), which is combined with a multiplier bootstrap to produce simultaneously valid time-varying P-values. Theorems 4 and 5 then apply Benjamini–Yekutieli and Benjamini–Hochberg procedures to these dependent P-values and establish sup_t FDR control. The paper also contains numerical simulations and applications to EEG and financial data.","tokens_in":30906,"tokens_out":6120,"duration_ms":60987,"significance":"If the main theorems are correct, the paper makes a useful contribution: it extends earlier FWER-based inference for time-varying correlation networks (Bai and Wu, 2025) to FDR control, under a broad model class that allows non-Gaussianity, non-stationarity, and abrupt mean changes. The proposed procedure is clearly described, and the simulation and real-data analyses are informative. A particular strength is that the final FDR-control argument is transparent and uses standard B-Y/B-H calculations. However, the central theoretical claim is conditional on a high-dimensional Gaussian approximation (Lemma 3) whose proof is only sketched by invoking an author-overlapping preprint, and on Lemma 1 whose proof is omitted. These are load-bearing technical premises, not peripheral details.","major_comments":[{"comment":"The arXiv abstract defines AuFDR as the expectation of the L_r-norm of the FDP process with r allowed to diverge, but this quantity does not appear anywhere in the main text. Theorems 4 and 5 instead prove sup_{t∈[b,1-b]} [FDR(t) − 2α|H0(t)|/(p(p−1))] ≤ 0. These are different criteria: control of the time-wise supremum of pointwise FDR is not the same as control of an L_r norm (even for diverging r). The authors should either define AuFDR in the body and prove a theorem for it, or remove the AuFDR claim from the abstract/title. This is a mismatch in the paper's advertised central contribution.","section":"Abstract vs. §1 and §4 (Theorems 4–5)"},{"comment":"Lemma 3 is the key Gaussian approximation used in Theorem 1, and hence in Theorems 3–5. Its proof is not self-contained: equation (28) is asserted to follow from Propositions 6–7 of Wu et al. (2024) with “minor modifications” after a “very careful investigation”, but two of the modifications are explicitly non-minor: Lemma 2(iii) gives δ_q(H_{i,l,Ξ},r)=O(χ^{r−h}) rather than O(χ^r), and the tail condition is sub-Weibull rather than exponential. These changes introduce the terms (φ^q n p^* χ^{(M−h)q})^{1/(1+q)} and exp(−tM_x^κ) in (28), but the subsequent optimization in M, M_x, M_y, φ, β that yields the claimed O((nb)^{−1/8}(log n)^4) bound is not shown. There is also no explicit accounting of how the dimension p enters the final sup over hyperrectangles. Since the FDR theorems require p^2(θ_n^{(1)} + θ_n^{(2)}) → 0, an uncounted polynomial-in-p factor in Lemma 3 would invalidate the mai","section":"Appendix E.3, Lemma 3"},{"comment":"Lemma 1 is used in the proof of Theorem 1 to pass from the discretized maximum to the supremum over t and to compare the original statistic with the Gaussian proxy (see equations (20)–(21) and subsequent steps). The lemma is stated with the sentence “For simplicity we omit the proof” and a citation to Lemma S1 of Dette and Wu (2021). This is not adequate for a lemma on which the main theorem depends. The authors should provide a full proof or state the exact Dette–Wu lemma with the precise adaptation and verify its hypotheses in this setting.","section":"Appendix E.1, Lemma 1"}],"minor_comments":[{"comment":"There are two different titles/abstracts: the arXiv version advertises “Asymptotic Uniform False Discovery Rate Control for Inference of Time-varying Correlations” and an L_r-based AuFDR, while the main text is titled “Learning Time-Varying Correlation Networks with FDR Control via Time-varying P-values” and uses a sup-FDR criterion. Please harmonize these.","section":"Throughout"},{"comment":"Typo: “investiated” should be “investigated”; elsewhere “guaranties” should be “guarantees” and “provides theoretical guarantees for the B-H and B-Y procedure” is repeated with a grammatical error in the sentence before Theorem 4.","section":"§1"},{"comment":"The statement of Lemma 1 begins with “for every y∈R^{p*} and δ>0” followed by a supremum over y; the first “for every y” is redundant or misstated. Please clarify the quantifiers.","section":"Appendix E.1, Lemma 1"},{"comment":"The denominator for the bootstrap statistic is √(2w⌈nb⌉), where b = max_{i,l} b_{i,l}, while the numerator is computed with a sum over 2⌈nb_{i,l}⌉−w. If the use of the global b is intentional due to the scaling c_{i,l} in (8), please state this explicitly; otherwise this appears to be a typo.","section":"Algorithm 1, step 6"},{"comment":"The text says Assumption 2 is satisfied for sufficiently large q if ω≍n^{2/5}; the parameter ω is not defined. It is presumably the block size w. Please correct the notation.","section":"§4, Assumption 2 paragraph"}],"recommendation":"major_revision","confidential_remarks":"The manuscript’s central theorem depends crucially on Lemma 3, an extension of Propositions 6–7 of Wu et al. (2024), an author-overlapping preprint. The modifications described are not trivial, and the presented proof is a sketch. I would recommend that the editor require a complete, self-contained proof of Lemma 3 (and Lemma 1) before publication, even if it adds substantial supplementary material. The abstract/main-text inconsistency about AuFDR should also be resolved editorially."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThis paper is a genuine step beyond Bai and Wu (2025): it moves from pointwise FWER to uniform-in-time FDR control for time-varying correlation networks, under non-stationarity, non-Gaussianity, and mean jumps. The difference-based estimator, the multiplier bootstrap, and the B-H/B-Y wrappers are all sensible, and the simulations support the empirical claims. The real-data sections are a nice addition.\n\nThe soft spot is exactly where the stress-test puts it. The whole edifice—Theorems 4 and 5—rests on simultaneous validity of the bootstrap P-values, which comes from Theorem 1, which uses Lemma 3: a hyperrectangular Gaussian approximation for the product sequence. That lemma is not proved in this paper. It is asserted to follow from Propositions 6–7 of Wu et al. (2024) with 'minor modifications' after a 'very careful investigation.' The modifications are not minor: the innovations are lag-h differences, so Lemma 2(iii) gives δ_q(H_{i,l,Ξ}, r) = O(χ^{r-h}) rather than O(χ^r), and the tail condition is sub-Weibull, not exponential. The error bound (28) picks up extra terms, and the final O((nb)^{-1/8}(log n)^4) rate is claimed without explicit p-dependence. If a hidden polynomial-in-p factor appears, the condition p^2(θ_n^{(1)} + θ_n^{(2)}) → 0 in Theorems 4–5 fails. Lemma 1, also needed for Theorem 1, has its proof omitted. These are not peripheral details; they are the bridge from the bootstrap distribution to P-value uniformity.\n\nThere is also a smaller mismatch: the abstract defines AuFDR as the expectation of an L_r norm of the FDP process with r diverging, but the body never uses that definition. The body works with sup_t [FDR(t) − α·2|H0(t)|/(p(p−1))] ≤ 0. That needs to be reconciled.\n\nGiven all that, I would not desk-reject this. The derivations that are shown are coherent, the FDR argument given valid P-values is standard, and the problem is practically important. I would send it to a serious referee, with the request that Lemma 3 be either fully proved or stated precisely with all constants and explicit p-dependence, and that Lemma 1 get a proof. The paper is for people working on high-dimensional time-series inference and multiple testing; they will get value from the methodological framework even while the theoretical guarantee is pending.\n\nRecommendation: engage, but demand the missing proofs before taking the uniform FDR claim at face value.","headline":"Genuine extension of the FWER work to uniform FDR control, but the load-bearing Gaussian approximation lemma is not actually proved here.","tokens_in":31384,"tokens_out":2766,"would_cite":false,"duration_ms":25886,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H15","62M10","62F40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that false discovery rate control can be enforced uniformly over the entire time range when constructing time-varying correlation networks from high-dimensional, locally stationary, non-Gaussian time series with jumps in t","keywords":["time-varying correlation network","false discovery rate","locally stationary time series","multiplier bootstrap","Gaussian approximation","non-stationary time series","change points","high-dimensional inference"],"falsifier":"Simulate a locally stationary Gaussian process with zero true correlations at all times and known jump locations, run Algorithm 1 with the B-Y procedure at level α over a dense grid of t, and check whether the empirical sup_t FDP exceeds α (or the paper's bound) as n grows. A more direct check is to compare the empirical distribution of the bootstrap maximum deviation to the Gaussian approximation in the proof of Lemma 3 for a simple AR(1) model; if the Kolmogorov distance decays slower than (nb)^{-1/8}(log n)^4, the proof fails.","tokens_in":30330,"feed_emoji":"📊","tokens_out":4380,"duration_ms":45618,"temperature":0.7,"pith_summary":"The paper aims to show that false discovery rate (FDR) control can be achieved simultaneously across all time points when learning time-varying correlation networks from high-dimensional, locally stationary, possibly non-Gaussian time series with jumps in their means. It constructs time-varying P-values for every pair of series at every time point using a multiplier bootstrap applied to difference-based estimates of the time-varying correlation. It then proves that applying the Benjamini–Yekutieli procedure (or the Benjamini–Hochberg procedure under a positive-dependence condition) keeps the FDR below the target level uniformly over time. If correct, this gives practitioners a principled way to threshold correlation networks at arbitrary time points without losing error control.","feed_headline":"Uniform false-discovery control for time-varying correlation networks","feed_subtitle":"Bootstrapped time-varying P-values keep expected false edges below target at every moment.","key_machinery":"The load-bearing object is the difference-based local-linear estimator of the time-varying covariance, reduced to a weighted moving average of an innovation process Ξ_{j,i,l}. A new hyperrectangle Gaussian approximation shows that the joint distribution of the standardized deviations can be replaced by a Gaussian vector with the same autocovariance structure; the multiplier bootstrap replicates that Gaussian vector. The resulting time-varying P-values feed directly into the B-H and B-Y thresholds.","core_discovery":"The central claim is that the vector of bootstrap statistics across all coordinate pairs and time points is asymptotically distributed like a high-dimensional Gaussian process whose hyperrectangle probabilities match those of the standardized correlation estimates, making the time-varying P-values simultaneously valid. Consequently, Theorem 4 states that for the Benjamini–Yekutieli procedure, lim_{n→∞} sup_{t∈[b,1−b]} [FDR(t) − 2α|H0(t)|/(p(p−1))] ≤ 0, and Theorem 5 gives the same for the Benjamini–Hochberg procedure under a PRDS assumption. The guarantees hold with the dimension p growing polynomially in the sample size n.","pith_inferences":["Editorial inference: the uniform-in-time guarantee means the network can be treated as a continuous object; one could in principle threshold the estimated correlation surface at every time and report the resulting dynamic graph without re-adjusting α.","Editorial inference: the bootstrap P-value construction is modular; the same hyperrectangle Gaussian approximation should extend to lagged (directed) correlation networks, which the paper only sketches, and possibly to partial correlations, a direction the authors list as future work.","Editorial inference: a natural stress test is to run the method on data with all null correlations and record sup_t FDP; if the B-Y bound is loose, one could improve power by replacing the log(p∗)+γ penalty with a data-dependent dependence-aware threshold."],"forward_implications":["At any time t, the network edge set can be read off the B-Y threshold with a uniform guarantee that the expected proportion of false edges is at most α times the null proportion, up to asymptotically negligible terms.","Because the guarantee is uniform in t, an analyst can scan the whole time interval for edges without a multiple-comparison penalty that grows with the number of time points examined.","The B-H variant, valid under positive dependence, gives FDP close to the target with lower false negative proportion; B-Y is more conservative but safe under arbitrary dependence.","True edges are detected with probability tending to 1: P-values for alternatives tend to 0.","The framework tolerates non-Gaussian, nonlinear, locally stationary errors with jumps in the mean, so it applies to EEG and financial returns."],"fun_headline_variants":["Uniform FDR control for nonstationary correlation networks","Time-varying P-values for uniform false discovery control","Bootstrap-based FDR control for time-varying correlations","Asymptotic guarantees on false discoveries over time"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole argument leans on an unproven hyperrectangular Gaussian approximation rate (Lemma 3, adapted from a prior work with 'minor modifications', and Lemma 1 stated without proof); if that rate fails, the simultaneous validity of the P-values—and with it both FDR theorems—collapses.","fun_headline_variants_meta":{"raw":{"variants":["Uniform FDR control for nonstationary correlation networks","Time-varying P-values for uniform false discovery control","Bootstrap-based FDR control for time-varying correlations","Asymptotic guarantees on false discoveries over time"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000437,"raw_usage":{"total_tokens":2052,"prompt_tokens":733,"completion_tokens":1319,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":1270}},"tokens_in":477,"tokens_out":1319,"duration_ms":11201,"temperature":1.0,"reasoning_tokens":1270,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T17:06:51.642370+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a locally stationary Gaussian process with zero true correlations at all times and known jump locations, run Algorithm 1 with the B-Y procedure at level α over a dense grid of t, and check whether the empirical sup_t FDP exceeds α (or the paper's bound) as n grows. A more direct check is to compare the empirical distribution of the bootstrap maximum deviation to the Gaussian approximation in the proof of Lemma 3 for a simple AR(1) model; if the Kolmogorov distance decays slower than (nb)^{-1/8}(log n)^4, the proof fails.","supporting_citations":[],"review_version":1}