{"id":"db5a0a1a-cc2a-4619-b2c1-abbcd120367a","arxiv_id":"2506.05112","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Replacing the additive penalty in multiscale scan tests by a multiplicative weighting yields critical values that are asymptotically valid for sub-Gaussian noise, based on a new thresholded weak convergence result.","lead":"Multiscale scan statistics usually need Gaussian noise or known tail bounds, and existing fixes lose power on short signals. This paper proposes a multiplicatively weighted statistic whose asymptotics hold for general sub-Gaussian errors, via a new 'thresholded weak convergence' phenomenon, and applies it to changepoint detection and the April 2025 Iberian blackout.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The feasibility claim fails at any preset alpha without an estimate or bound for the sub-Gaussian constant C_eta: Theorem 3.1 requires q_alpha > C_eta, Proposition 2.7 shows that for every finite alpha there exist unit-variance sub-Gaussian laws with asymptotic size 1, and no procedure in the…","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the unknown sub-Gaussian constant C_eta and the condition q_alpha > C_eta. My stress-test confirms this is the single most consequential gap in the paper's central claim. The probabilistic core of the paper, thresholded weak convergence in the upper tail, appears to be internally sound; the issue is that the statistical procedure derived from it is 'feasible' only relative to an unknown constant that determines whether a chosen significance level is in the valid regime. Proposition 2.7 makes this precise: for any finite critical value there exists a unit-variance sub-Gaussian law for which the test statistic stays above that value with probability tending to one, so no fixed alpha is uniformly safe over the paper's class of errors. The simulations in Table 3 are internally consistent with this mechanism, not a finite-sample anomaly. The proof-of-boundedness issue flagged by the reader (the apparent need for C < 2 in Theorem 2.2) is reparable by replacing the constant 2 with an arbitrary t0 > C in the argument, so it is not the load-bearing concern. Because the reader already issued a CONDITIONAL verdict based on the same weakness, my stress-test does not change the verdict. The paper would need either an estimable upper bound for C_eta, an adaptive significance level construction, or a clear restriction of the feasibility claim to levels alpha for which the condition can be verified, before the multiscale test can be used at preset significance levels.","tokens_in":32118,"tokens_out":9856,"duration_ms":113384,"concrete_test":"Simulate the signal-discovery test under the mixture noise 1/2 N(0,2) + 1/2 delta_0 with n = 10^6, using the full-grid critical value q_alpha = 2.384 at alpha = 10% (so q_alpha > C_eta ~ 2.12). If the empirical size converges to about 10% while the dyadic-grid version with q_alpha = 1.907 remains inflated, this confirms that the threshold condition is the operative constraint. Then estimate C_eta from the residuals (e.g., via the empirical MGF bound sup_r E exp(r eta)/r^2) and check whether choosing alpha so that q_alpha > \\hat C_eta restores size control; if it does not, the feasibility claim remains unresolved because the required constant is not practically accessible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central statistical claim is that the multiplicatively weighted statistic T*_n yields a feasible multiscale test for non-Gaussian sub-Gaussian noise, agnostic of the tail bound. This claim is not supported at any fixed significance level. Corollary 2.6 and Theorem 3.1 establish size control only if q_alpha > C_eta, where C_eta is the MGF constant in E exp(r eta_t) <= exp(r^2/C_eta^2). The paper estimates only the variance sigma^2 (via bsigma_n); it provides no estimator, upper bound, or diagnostic for C_eta. Consequently, a practitioner cannot know whether a chosen alpha, e.g. 5%, satisfies the validity condition. This is not a minor finite-sample issue: Proposition 2.7 shows that for any finite T there is a unit-variance, centered, sub-Gaussian law with liminf P(|S_n|_{rho2} >= T) = 1. Taking T = q_alpha, for every preset alpha there exists a distribution in the paper's class for which the test has asymptotic size 1. The simulations confirm the mechanism: for mixture (c) on the dyadic grid, q_alpha = 1.907 at alpha = 10% is below C_eta ~ 2.12 and the size is 46% even at n = 50000, while at alpha = 0.1%, q_alpha = 2.631 > C_eta and the size is approximately nominal. Thus the method's validity hinges on an unverifiable distributional constant, undermining the advertised statistical feasibility.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a notion of thresholded weak convergence and proves that interpolated partial sum processes of sub-Gaussian innovations satisfy such convergence for the H\\\"older-type seminorm with critical modulus \\rho_2(h)=\\sqrt{h\\log(e/h)}, even though classical weak convergence in the space C^{\\rho_2} fails. On this basis the authors construct multiplicatively weighted multiscale tests for signal detection, goodness-of-fit, and multiple changepoint localization, extend the results to locally stationary nonlinear time series through a physical-dependence concentration inequality and a bootstrap, and illustrate the methodology with simulations and an analysis of the April 2025 Iberian power-grid blackout.","tokens_in":32366,"tokens_out":23708,"duration_ms":286802,"significance":"The concept of thresholded weak convergence at the critical H\\\"older modulus is novel and potentially useful beyond the present applications; the paper correctly identifies a genuine boundary phenomenon in Donsker-type theorems. The statistical motivation for replacing the additive multiscale penalty by a multiplicative weight is well argued, and the paper contains a substantial simulation study and a real-data application. The proofs are conventional and largely reproducible in structure. However, several quantitative claims that are load-bearing for the advertised feasibility are not correct as stated: the threshold in the main convergence theorem appears to be off by a factor, the dependent-data concentration inequality omits a leading term, and the condition for size control depends on an unknown and unestimated sub-Gaussian constant. These issues require substantial revision before the results can be accepted.","major_comments":[{"comment":"The stated threshold is not the correct one for the sub-Gaussian constant. For innovations satisfying E exp(r\\eta_t) \\le \\exp(r^2/C^2), the tail of a normalized sum over an interval of length h has the form P(|W_n(u)-W_n(v)|/\\rho_2(h)>t) \\le 2\\exp(-C^2 t^2 \\log(1/h)/4) = 2 h^{C^2 t^2/4}. Thus in condition (T) of Theorem 2.2 one has \\kappa(t)=C^2 t^2/4, and \\kappa(t)>1 requires t>2/C, not t>C. Corollary 2.6 and Theorem 3.1 therefore need the condition q_\\alpha > 2/C_\\eta rather than q_\\alpha > C_\\eta. The simulation evidence in Table 3 is consistent with the corrected threshold: for mixture (c), C_\\eta=1, 2/C_\\eta=2, q_{10\\%}=1.907<2 and the size does not approach 10\\%, whereas q_{0.1\\%}=3.316>2 and the size is near nominal. As written, Corollary 2.6 is quantitatively false for, e.g., \\eta_t \\sim \\frac12 N(0,2)+\\frac12\\delta_0.","section":"Corollary 2.6 and Theorem 3.1"},{"comment":"The sub-Gaussian concentration inequality for dependent data is missing the leading term. The statement bounds \\|\\sum_{t=1}^n w_t\\eta_t\\|_{\\psi_2} by K\\sqrt{\\sum_t |w_t|^2}\\,\\sum_{j=1}^\\infty \\sqrt{j}\\,\\delta_{\\psi_2}(j). For iid innovations, \\delta_{\\psi_2}(j)=0 for all j\\ge1, so the right-hand side is zero while the left-hand side is generally positive and of order \\sqrt{n}. The proof telescopes from S_{n,0}=\\sum_t w_t E(\\eta_t|\\epsilon_t), which is not S_n; the j=0 term carries the main contribution and is omitted from the series. Corollary 2.10 relies on Theorem 2.8, so the dependent-data extension is not supported as stated. The series should include a j=0 term (e.g., \\delta_{\\psi_2}(0)=\\sup_t\\|\\eta_t\\|_{\\psi_2}) and the proof adjusted accordingly.","section":"Theorem 2.8"},{"comment":"The size guarantee depends on the unverifiable condition q_\\alpha > 2/C_\\eta (or, as stated in the paper, q_\\alpha > C_\\eta). The paper estimates only the variance \\sigma^2 and provides no estimator, diagnostic, or upper bound for C_\\eta. Proposition 2.7 shows that for any finite T there exists a unit-variance, centered, sub-Gaussian law with \\liminf_{n\\to\\infty} P(|\\tilde S_n|_{\\rho_2}\\ge T)=1; taking T=q_\\alpha, for every preset \\alpha there is a distribution in the stated class for which the test has asymptotic size 1. The abstract's claim of a 'feasible multiscale test' that is 'agnostic of the exact tail bound' is therefore overstated: the procedure is feasible only for significance levels below an unknown \\alpha_0. The authors should either provide a way to estimate or bound the threshold or carefully delimit the feasibility claim in the abstract and in Sections 1 and 3.1.","section":"Theorem 3.1 and Proposition 2.7"},{"comment":"The stochastic boundedness argument contains a gap. The proof uses Q(t/3,1/N) \\le Q(2,1/N) and then chooses N so that Q(2,1/N)<\\varepsilon/2. This requires the number 2 to exceed the threshold C in condition (T). The theorem does not assume C<2, and in the dependent-data setting C may exceed 2, so the chosen N need not exist. This step should be replaced, for example by letting t\\to\\infty for a suitable N using the fact that Q(t/3,1/N)\\to0 for t>C, together with a more careful control of the finite-dimensional term, or by an alternative tightness argument. As written, the proof of the first claim in (4) is incomplete.","section":"Appendix, proof of Theorem 2.2"}],"minor_comments":[{"comment":"The caption contains the typo 'Gassian'; it should read 'Gaussian'.","section":"Figure 1 caption"},{"comment":"The text states the intervals are declared at nominal significance level 1% (99% confidence), but the caption says 'significant at 95%'. These should be made consistent.","section":"Section 4.3 / Figure 3 caption"},{"comment":"The sentence 'the the threshold \\tau is the same' contains a duplicated 'the'.","section":"Section 3.5"},{"comment":"There are typos: 'loosing too much finite sample power' should be 'losing', and in Table 2 'comparsion' should be 'comparison'.","section":"Section 1 and Table 2"},{"comment":"The phrase 'A a sequence of real-valued random variables' should read 'A sequence of real-valued random variables'.","section":"Definition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a genuinely interesting probabilistic idea and a substantial statistical framework, but the threshold constant error in Corollary 2.6 and the missing leading term in Theorem 2.8 are load-bearing and must be corrected. The feasibility caveat around the unknown sub-Gaussian constant also needs to be addressed honestly. I believe these issues are fixable within the scope of a major revision, so I do not recommend rejection at this stage."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper introduces thresholded weak convergence and proves it at the critical Hölder modulus where Donsker's theorem fails. The upper tail of |S_n|_rho2 converges to that of Brownian motion, the lower tail does not, and Proposition 2.7 shows the threshold is real, not a proof artifact. That is a genuine new result, and the arguments look largely correct.\n\nThe statistics built on it are clever. The multiplicatively weighted statistic avoids the additive penalty that breaks under non-Gaussian noise, and the paper delivers size control for sub-Gaussian errors, plus extensions to dependent nonstationary series via a new concentration inequality (Theorem 2.8). The changepoint and goodness-of-fit applications are reasonable, and the simulations are unusually honest—they report size inflation at 10% for a moderately heavy-tailed mixture, which matches their theory.\n\nThe main soft spot is the feasibility claim. Size control requires q_alpha > C_eta, where C_eta is the sub-Gaussian MGF constant. The paper gives no estimator or bound for C_eta. Proposition 2.7 implies that for any fixed alpha, there is a sub-Gaussian law in the class with asymptotic size 1. So the procedure is not uniformly valid over the class; it is valid only for distributions whose tail constant is small enough. The paper states the condition, but the abstract's 'feasible' overstates the practical reach. This is not fatal to the probabilistic core, but it should be addressed before the method is presented as turnkey. A natural fix is an adaptive alpha or a data-driven check on C_eta.\n\nThere is also a minor proof gap in Theorem 2.2: the stochastic boundedness step uses Q(2,1/N) as if 2 exceeded the threshold C, which is not guaranteed for dependent data. It is reparable by replacing 2 with a threshold above C. Not a substantive concern.\n\nThis paper is for anyone working on multiscale inference, scan statistics, or functional CLTs in Hölder spaces. The thresholded convergence result will likely be cited. It deserves a serious referee—I would send it out, expecting the authors to clarify the alpha condition and patch the proof gap.","headline":"A genuine probabilistic novelty—thresholded weak convergence at the critical Hölder modulus—with a statistical wrapper whose 'feasibility' claim overreaches because the alpha condition depends on an unknown tail constant.","tokens_in":33040,"tokens_out":4395,"would_cite":true,"duration_ms":47442,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F17","62G10","62G20","62M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the multiplicatively weighted multiscale scan statistic is asymptotically pivotal under sub-Gaussian noise, through a new thresholded form of weak convergence that salvages distributional limits exactly where…","keywords":["multiscale testing","thresholded weak convergence","scan statistic","Hölder spaces","sub-Gaussian errors","changepoint detection","goodness-of-fit testing","locally stationary time series"],"falsifier":"Simulate the signal discovery null with $n=50{,}000$ iid draws from the mixture $\\frac12 N(0,2)+\\frac12\\delta_0$ and record the empirical rejection rate at nominal 10% using the Brownian-motion critical value; the paper's Table 3 already reports roughly 46% rejection, showing that the claimed size control fails exactly when $q_\\alpha \\le C_\\eta$.","tokens_in":31765,"feed_emoji":"📊","tokens_out":4981,"duration_ms":55856,"temperature":0.7,"pith_summary":"This paper sets out to fix a gap in multiscale testing: the standard multiscale scan statistic, which subtracts a scale-dependent penalty, only works for Gaussian noise and collapses for slightly heavier sub-Gaussian errors. The authors propose replacing the additive penalty by a multiplicative weight, giving the statistic $T^*_n = \\max_I T_n(I)/\\sqrt{\\log(en/|I|)}$. They prove this statistic is asymptotically pivotal under sub-Gaussian errors in a new thresholded sense: tail probabilities converge to those of a Hölder seminorm of Brownian motion, but only above a threshold tied to the unknown tail constant. Because critical values can be computed from Brownian motion once the noise variance is estimated, the test is feasible in practice, preserves optimal detection rates, and extends to nonstationary dependent data via bootstrap.","feed_headline":"Thresholded limit rescues multiscale tests for non-Gaussian noise","feed_subtitle":"A pivotal scan statistic whose Brownian-motion critical values survive non-Gaussian, dependent, and nonstationary errors.","key_machinery":"The engine is thresholded weak convergence: a sequence of random variables $X_n$ converges beyond a threshold $\\tau$ to $X$ if $P(X_n>t)\\to P(X>t)$ for every continuity point $t>\\tau$. The sufficient criterion in Theorem 2.2 combines a sub-Gaussian tail bound on increments with a scaling property of the modulus, producing control of the Hölder seminorm in the upper tail despite the failure of tightness in the full space $C^{\\rho_2}$. This turns Donsker's theorem at its edge of validity into a statistically usable statement.","core_discovery":"The central claim is that the multiplicatively weighted scan statistic admits a distributional limit even though the usual functional central limit theorem fails at the critical modulus of continuity $\\rho_2(h)=\\sqrt{h\\log(e/h)}$. Concretely, for iid centered sub-Gaussian errors with unit variance, $|\\tilde S_n|_{\\rho_2}/\\sigma$ converges in tail to $|B|_{\\rho_2}$ for thresholds above the tail constant $C_\\eta$, where $B$ is standard Brownian motion. The paper shows that the convergence genuinely does not hold below that threshold, so the thresholding is a real property of the statistic rather than an artifact of the proof.","pith_inferences":["Extension: the same thresholded-limit mechanism should transfer to other sup-functionals over Hölder seminorms, such as density or deconvolution settings, wherever sub-Gaussian tail bounds replace Gaussianity.","Extension: the unknown $C_\\eta$ threshold suggests a practical diagnostic, namely estimating the effective tail constant from residuals and checking $q_\\alpha > C_\\eta$ before trusting a chosen level; the paper does not implement such a check.","Extension: a data-driven choice of the modulus parameter $a$ in $\\rho_{2,a}(h)=\\sqrt{h(a+\\log(e/h))}$ could trade short-signal power against robustness without obviously losing the threshold guarantee, and that trade-off is testable by simulation.","Extension: for changepoint inference the paper leaves open whether the sharper Gaussian localization rate $O(1/\\delta_k^2)$ can be reached for non-Gaussian errors, which is a natural next target."],"forward_implications":["For iid sub-Gaussian noise, the test $T^*_n$ with Brownian-motion critical values has asymptotic size at most $\\alpha$ whenever the quantile $q_\\alpha$ exceeds the unknown tail constant $C_\\eta$.","A signal with amplitude $\\mu_n$ and length $\\ell_n$ is consistently detected whenever $\\mu_n^2\\ell_n \\gg \\log(en/\\ell_n)$, the same optimal rate as the Gaussian multiscale statistic.","Goodness-of-fit testing and multiple changepoint localization inherit the same validity, with changepoint localization rates matching statistical lower bounds.","A multiplier bootstrap based on block sums gives critical values for nonstationary, locally stationary dependent errors, provided the block length and cutoff satisfy stated rates.","Sparse dyadic and Rivera-Walther grids reduce computation to $O(n)$ or $O(n\\log n)$ without sacrificing the asymptotic detection guarantees."],"supporting_citations":[{"why":"Supplies the additive multiscale statistic and the optimal detection benchmark that the multiplicative weighting is designed to replace.","marker":"Dümbgen & Spokoiny (2001)"},{"why":"Provides the classical Donsker theorem whose failure at the critical Hölder modulus motivates the thresholded convergence concept.","marker":"Billingsley (1999)"},{"why":"Establishes the tradition of invariance principles in Hölder spaces that this paper extends to the critical case.","marker":"Lamperti (1962)"},{"why":"Formulates conditions for Hölderian invariance principles that stop short of the critical modulus $\\rho_2$, defining the edge of applicability.","marker":"Rackauskas & Suquet (2004c)"},{"why":"Gives the Lévy modulus of continuity of Brownian motion used to identify the threshold and the limiting distribution.","marker":"Schilling (2021)"},{"why":"Provides the multiscale changepoint procedure whose interval-length lower bound causes suboptimal power that the new method avoids.","marker":"Frick et al. (2014)"},{"why":"Supplies the narrowest significance pursuit benchmark for changepoint localization against which the new intervals are compared.","marker":"Fryzlewicz (2024a)"},{"why":"Defines the physical dependence measure used in the sub-Gaussian concentration bound for dependent data.","marker":"Wu (2005)"}],"fun_headline_variants":["Thresholded limit fixes multiscale tests beyond Donsker","Scan statistics get feasible multiscale tests via thresholded limit","Donsker fails, but multiscale tests still work via thresholded convergence","New limit overcomes Donsker failure for multiscale scan statistics","Multiscale testing without Donsker: thresholded weak convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the assumption that errors are sub-Gaussian with a finite constant $C_\\eta$ that is never estimated: the size guarantee holds only when the chosen significance level has critical value $q_\\alpha > C_\\eta$, so in practice a user cannot verify that the level is small enough.","fun_headline_variants_meta":{"raw":{"variants":["Thresholded limit fixes multiscale tests beyond Donsker","Scan statistics get feasible multiscale tests via thresholded limit","Donsker fails, but multiscale tests still work via thresholded convergence","New limit overcomes Donsker failure for multiscale scan statistics","Multiscale testing without Donsker: thresholded weak convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00029,"raw_usage":{"total_tokens":1673,"prompt_tokens":899,"completion_tokens":774,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":691}},"tokens_in":515,"tokens_out":774,"duration_ms":7960,"temperature":1.0,"reasoning_tokens":691,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:25:51.685626+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the signal discovery null with $n=50{,}000$ iid draws from the mixture $\\frac12 N(0,2)+\\frac12\\delta_0$ and record the empirical rejection rate at nominal 10% using the Brownian-motion critical value; the paper's Table 3 already reports roughly 46% rejection, showing that the claimed size control fails exactly when $q_\\alpha \\le C_\\eta$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical Donsker theorem whose failure at the critical Hölder modulus motivates the thresholded convergence concept."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the tradition of invariance principles in Hölder spaces that this paper extends to the critical case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Lévy modulus of continuity of Brownian motion used to identify the threshold and the limiting distribution."},{"cited_title":", Munk, A","cited_arxiv_id":null,"evidence_quote":"Provides the multiscale changepoint procedure whose interval-length lower bound causes suboptimal power that the new method avoids."}],"review_version":1}