{"id":"6de88891-2f58-493a-82b8-6f3dcb32a08a","arxiv_id":"2607.13173","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In sliding-window statistics on i.i.d. data, the genuinely new information gained by increasing the window size is exactly the eigenvalue-one component of the covariance operator.","lead":"For overlapping sliding-window statistics on i.i.d. sequences, this paper characterizes the new information gained by enlarging the window as the eigenvalue-one component of the covariance operator. It provides explicit projection formulas for correlation and trend statistics, variance normalizations, and a scale-selection index, with simulation evidence.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.2's asymptotic variance for the incremental correlation statistic is off by a factor of two, so the paper's normalization machinery is wrong as printed.","rationale":"The reader's weakest assumption was the unproved external Theorem 2.3, which is indeed load-bearing for the structural claim. However, the theorem is cited from a published paper and the internal construction is consistent with its statement. The more decisive, checkable defect is the factor-of-two error in Proposition 4.2, which the reader also noted in the rationale but did not identify as the weakest assumption. This error is concrete, reproducible, and directly affects the normalization of the proposed correlation statistics, undermining the simulation results as printed. It does not overturn the central spectral decomposition, so it supports the reader's CONDITIONAL verdict rather than rejection. No ad hominem or theatrical language is needed; the arithmetic check is dispositive on the narrow point of normalization.","tokens_in":17574,"tokens_out":22847,"duration_ms":176818,"concrete_test":"Recompute Proposition 4.2 directly for k=4 with μ=0, σ^2=1. The proposed formula gives σ^2(g1)=1/2, but direct evaluation gives σ^2(g1)=E[(X1X4X2X3)^2]=1 because g1∈L1. More generally, expand E[S^2] with the corrected counting in Proposition 4.2: E[S^2]=C(k-2,2)α^2+2(k-2)C(k-3,2)αμ^2+6C(k-2,4)μ^4; the bracket in the published formula is exactly half of the resulting expression. This arithmetic check settles whether the normalization is usable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's practical payoff is explicit normalization of the constructed tests, and the most load-bearing defect is a concrete arithmetic error in Proposition 4.2. For g1 in L1, the spectral variance formula gives σ^2(g1)=||g1||^2. Let m=k-2 and S=Σ_{2≤i<j≤k-1} XiXj. A direct count gives E[S^2]=C(m,2)α^2+2m C(m-1,2)αμ^2+6C(m,4)μ^4, with α=μ^2+σ^2. Factoring out (k-2)(k-3)/4 yields [2α^2+4(k-4)αμ^2+(k-4)(k-5)μ^4], exactly twice the bracket printed in Proposition 4.2. Concretely, for k=4, μ=0, σ^2=1, g1=X1X2X3X4; Proposition 4.2 gives σ^2(g1)=1/2, while ||g1||^2=1. Since g1 is an eigenfunction with eigenvalue 1, the asymptotic variance cannot be 1/2. This error propagates to the normalization used in the simulations and to Proposition 4.8's stated σ^2(g1)=C(k-2,2) under N(0,1), which is inconsistent with Proposition 4.2. The central structural claim about the eigenvalue-one decomposition is not refuted by this, but the paper's claimed 'explicit variance formulas for normalization' are not correct as printed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a spectral framework for overlapping sliding-window statistics under an i.i.d. null. Each window statistic f is seen as an element of L^2(X^k), and the asymptotic variance is represented as <B_k f,f> for an overlap covariance operator B_k. The paper quotes a spectral theorem stating that B_k has eigenvalues 0,1,...,k, yielding an orthogonal decomposition H_k = L_0 ⊕ ... ⊕ L_k. The eigenvalue-one subspace L_1 is identified with the incremental dependence information introduced by enlarging the window, and an auxiliary decomposition is constructed to compute the projection onto L_1. This is applied to symmetric polynomial correlation statistics (elementary symmetric polynomials) and to a localized rank trend statistic K, with explicit variance formulas, a relative incremental index I_k(f), local asymptotic calculations, and simulations. The central claim is that every centered sliding-window statistic can be separated into overlap-level components and that the L_1 component captures exactly the new information added by increasing the window size.","tokens_in":17941,"tokens_out":9608,"duration_ms":82688,"significance":"If the framework is correct, it gives a systematic, constructive method for designing and normalizing overlapping-window tests, and it provides a natural quantitative notion of incremental information. The spectral decomposition and the explicit projection procedure are elegant and potentially useful for a broad class of statistics. The paper also contains concrete, checkable formulas, and the simulation study illustrates the predicted local detection scales. However, the practical payoff depends on the correctness of the explicit variance formulas, and Proposition 4.2 contains a concrete factor-of-two error. This is a load-bearing defect: the normalization of the proposed g_1 statistic is wrong as printed, and it is inconsistent with Proposition 4.8. The central structural claim about the eigenvalue-one decomposition is not refuted, and the error appears fixable, but the manuscript cannot be accepted in its current form.","major_comments":[{"comment":"The stated asymptotic variance for g_1 is off by a factor of 2. For k=4, μ=0, σ^2=1, g_1 = X1X2X3X4 and ||g_1||^2 = 1; since g_1 ∈ L_1, the spectral formula gives σ^2(g_1) = 1. The proposition prints (k-2)(k-3)/4 = 1/2. Tracing the proof, the count E[S^2] = C(k-2,2)α^2 + 2(k-2)C(k-3,2)αμ^2 + 6C(k-2,4)μ^4 is correct, but when it is multiplied by σ^4 the factor C(k-2,2) = (k-2)(k-3)/2 is replaced by (k-2)(k-3)/4, losing a factor of 2. The corrected formula is σ^2(g_1) = σ^4 C(k-2,2)[(μ^2+σ^2)^2 + 2(k-4)μ^2(μ^2+σ^2) + (k-4)(k-5)/2 μ^4].","section":"§4.2, Proposition 4.2"},{"comment":"Proposition 4.8 states that under N(0,1), σ^2(g_1) = C(k-2,2). This is the correct value (it equals ||g_1||^2), but it is inconsistent with Proposition 4.2, which gives half that value. The text says the identity follows from Proposition 4.2, which cannot be true. After correcting Proposition 4.2, the two statements become consistent, but as printed the normalization formulas contradict each other. This also calls into question the simulation claim of correct calibration, since the simulation section says the variance formula of Proposition 4.2 is used for normalization.","section":"§4.4, Proposition 4.8"}],"minor_comments":[{"comment":"Theorem 2.3, the spectral theorem for B_k, is quoted from the author's earlier paper [3] without proof. Citing a published result is acceptable, but since the entire framework depends on it, a short proof or a more self-contained statement in an appendix would strengthen the paper.","section":"§2.3, Theorem 2.3"},{"comment":"The final sentence 'Multiplying by σ^4 and replacing α gives the stated formula' hides the algebraic error described in the major comments. The simplification step should be written out explicitly.","section":"§4.2, proof of Proposition 4.2"},{"comment":"The simulation section asserts that all statistics are 'correctly calibrated under the null.' If the normalization used the printed Proposition 4.2, this cannot hold for g_1. The authors should either state which variance formula was actually used or re-run the simulations with the corrected formula.","section":"§5, simulations"},{"comment":"The statement that K_1 'does not appear to admit a standard Pitman-type local asymptotic regime' is based on simulations rather than a proof. The wording should distinguish an empirical observation from a proven fact.","section":"§4.2, Remark on K_1"},{"comment":"The adjoint P^* is used but not explicitly defined in the text. Defining it would improve readability.","section":"§2.2, operator definition"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of math.ST and the central decomposition idea is attractive. The main obstacle is the factor-of-two error in Proposition 4.2 and its downstream consequences for Proposition 4.8 and the simulation calibration claim. I see no reason to doubt the spectral decomposition framework itself, but the explicit variance formulas are a load-bearing part of the paper's applied contribution. A corrected version with the formula fixed and the simulations checked would be suitable for further consideration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The spectral framework is real: the asymptotic covariance operator of overlapping windows has integer spectrum, and the L1 (eigenvalue-one) component is exactly the orthogonal complement of the space of functions arising from smaller windows. So calling that component the incremental dependence information is mathematically grounded, not a slogan. Second, the paper contains a genuine, checkable error. Proposition 4.2's variance formula for the projected quartic statistic g1 is off by a factor of two. The stress-test count holds up: for k=4, l=4, N(0,1), g1 = x1x2x3x4 has norm squared 1 and eigenvalue 1, so its asymptotic variance is 1; the printed formula gives 1/2. The prefactor should be (k-2)(k-3)/2, not (k-2)(k-3)/4. Proposition 4.8 even states the correct value C(k-2,2) while citing Proposition 4.2, so the paper contradicts itself. This is an afternoon's fix, but as printed it breaks the normalization the simulations depend on.\n\nWhat is actually new: the constructive auxiliary decomposition (Theorem 3.1), the endpoint-interaction form of the L1 projection for symmetric polynomials (Proposition 4.1), and the relative incremental index with its k^-3 decay in both examples. The trend example is well done: only the longest lag survives in the incremental component K1, and the paper is honest that K1 has no Pitman regime under drift. Lemma 4.3 and the variance formula for the unprojected trend statistic check out.\n\nSoft spots, in proportion. The spectral theorem (Theorem 2.3) is quoted from the author's own earlier papers, not proved, so the framework is not self-contained; a referee should confirm the hypotheses on the sampling distribution carry over. The simulation section never states k and l for the correlation statistics and provides no code, so the finite-sample claims are not reproducible. There is also a tension worth flagging: the paper claims null calibration while printing a variance off by a factor of two; either the simulations used the corrected normalization or the calibration claim needs rechecking. The definition of incremental information as the L1 projection is a choice, but Proposition 3.6 gives it independent footing.\n\nWho this is for: statisticians who design overlapping serial tests or work on the spectral structure of overlapping block processes. It deserves a serious referee. My recommendation: send it out, ask the referee to verify the normalization formulas and demand the simulation parameters — conditional accept after those fixes.","headline":"A coherent and useful spectral framework for overlapping-window tests, with a real factor-of-two error in the printed normalization formula that needs fixing before the results work as stated.","tokens_in":18446,"tokens_out":16828,"would_cite":true,"duration_ms":137021,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62G20","62M10","60J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sliding-window tests: new information is the eigenvalue-one component","keywords":["sliding-window statistics","overlapping blocks","Markov chain covariance operator","spectral decomposition","incremental information","correlation detection","trend detection","local asymptotics"],"falsifier":"Compute the transition operator P for the overlapping-window chain on a small finite alphabet (e.g., binary states, k=3), form B_3 = I + P + P* + P^2 + (P*)^2, and check numerically whether its nonzero eigenvalues are exactly 1, 2, 3 with the claimed eigenspace decomposition; any mismatch would collapse the framework. Alternatively, pick a non-symmetric f, compute σ²(f) directly from the long-run covariance formula Var(f(Y_1)) + 2 Σ_{r=1}^{k-1} Cov(f(Y_1), f(Y_{1+r})), and compare it with Σ l‖f_l‖² obtained from the paper's auxiliary decomposition—disagreement would refute the spectral varianc","tokens_in":17400,"feed_emoji":"🪟","tokens_out":5753,"duration_ms":39582,"temperature":0.7,"pith_summary":"This paper establishes that for any centered sliding-window statistic computed from independent observations, the asymptotic variance is governed by a covariance operator whose eigenvalues are exactly the integers 0 through k. That spectral structure splits the space of window functions into orthogonal components tied to overlap level, and the component at eigenvalue 1 is precisely the information genuinely added when the window is enlarged from k-1 to k. The paper supplies a constructive algorithm—repeated integration over boundary variables—that extracts these components without knowing the operator's eigenvectors, plus explicit variance formulas for normalizing the resulting statistics. It then applies the machinery to correlation and trend detection, deriving the local detection scales (n^{-1/4} for quartic correlation statistics, n^{-1/2} for a rank-based trend statistic) and a relative incremental index that declines like k^{-3} in the worked examples. A careful reader would care because the framework turns window-size choice from an ad hoc tuning decision into a question with a spectral answer.","feed_headline":"Sliding-window tests: new information is the eigenvalue-one component","feed_subtitle":"A spectral decomposition yields variance formulas, scale selection, and local detection rates for window statistics.","key_machinery":"The covariance operator B_k = I + P + P* + ... + P^{k-1} + (P*)^{k-1}, where P is the transition operator of the overlapping-window Markov chain Y_i = (X_i,...,X_{i+k-1}); its integer spectrum and eigenspaces L_l are the backbone of the decomposition. The auxiliary spaces S_l, consisting of sums of l functions supported on consecutive blocks of length k-l+1 with side-centering conditions, provide a constructive route to the eigenspaces: the paper's main algorithm (Theorem 3.1) obtains s_l ∈ S_l by telescoping marginal integrals of f, and for symmetric f the s_l are automatically eigenfunctions. The identity L_1 = W_{k-1}^⊥, characterized by vanishing conditional expectations on the first and","core_discovery":"The central claim is that the covariance operator B_k of the overlapping-window Markov chain has spectrum {0,1,...,k}, so every centered window function f decomposes uniquely as f = f_0 + f_1 + ... + f_k with f_l in the eigenvalue-l eigenspace L_l, and the asymptotic variance equals sum_{l=1}^k l ||f_l||^2. The paper identifies the incremental information introduced by enlarging the window to size k as the projection of f onto L_1, and proves that L_1 is exactly the orthogonal complement of W_{k-1}, the space of functions representable from k-1 consecutive observations; equivalently, f is incremental iff its left and right conditional expectations vanish. For symmetric polynomial correlation","pith_inferences":["If the spectral decomposition is as clean as claimed, the same L_1-vs-W_{k-1} dichotomy could be used to build pure incremental tests for long-range lags without contamination from lower-span structure; the endpoint-interaction form gives a template that could be adapted to other dependence alternatives.","The paper's own remark that the projected trend statistic K_1 lacks a Pitman regime suggests a general phenomenon: isolating the incremental component removes the aggregation that produces standard local asymptotics, so pure incremental statistics may need different distributional tools; this is an open question the author flags.","The relative incremental index, though introduced for comparing nested window sizes, could equally serve as a metric for comparing different statistics at the same window, and its monotone decay in the examples hints at a broader principle about how information saturates with window length; a general theorem on decay rates is left as future work."],"forward_implications":["Every centered window statistic can be normalized by the variance formula σ²(f) = Σ_{l=1}^k l‖f_l‖², and the components f_l can be computed by the paper's constructive marginal-integration algorithm.","The incremental part f_inc = Proj_{L_1} f isolates the dependence structure that first appears at span k; statistics built purely from endpoint interactions (x_1-μ)(x_k-μ) g(x_2,...,x_{k-1}) lie entirely in L_1 and need no projection.","The relative incremental index I_k(f) = ‖f_inc‖²/σ²(f) gives a window-size selection rule; for both the polynomial correlation family and the rank-based trend family it decreases in k, asymptotically as k^{-3}, so beyond a certain scale larger windows add little genuinely new relative information.","Local detection rates follow from the lowest-order nonvanishing term of the mean under the alternative: the quartic correlation statistics are second-order (local scale n^{-1/4}), while the trend statistic K is first-order (scale n^{-1/2}), and simulations confirm these scalings."],"fun_headline_variants":["Sliding-window new info: the L1 component","Overlapping windows: eigenvalue-1 eigenspace holds the new signal","Window tests: spectral decomposition gives incremental part","How much new dependence? Look at the eigen-1 projection"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything rests on Theorem 2.3, quoted without proof from an earlier paper, that the covariance operator B_k has eigenvalues exactly 0,1,...,k and that each auxiliary space S_l splits as the eigenspace L_l plus a kernel component.","fun_headline_variants_meta":{"raw":{"variants":["Sliding-window new info: the L1 component","Overlapping windows: eigenvalue-1 eigenspace holds the new signal","Window tests: spectral decomposition gives incremental part","How much new dependence? Look at the eigen-1 projection"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001019,"raw_usage":{"total_tokens":4142,"prompt_tokens":753,"completion_tokens":3389,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":3330}},"tokens_in":497,"tokens_out":3389,"duration_ms":113412,"temperature":1.0,"reasoning_tokens":3330,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T05:58:55.124248+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the transition operator P for the overlapping-window chain on a small finite alphabet (e.g., binary states, k=3), form B_3 = I + P + P* + P^2 + (P*)^2, and check numerically whether its nonzero eigenvalues are exactly 1, 2, 3 with the claimed eigenspace decomposition; any mismatch would collapse the framework. Alternatively, pick a non-symmetric f, compute σ²(f) directly from the long-run covariance formula Var(f(Y_1)) + 2 Σ_{r=1}^{k-1} Cov(f(Y_1), f(Y_{1+r})), and compare it with Σ l‖f_l‖² obtained from the paper's auxiliary decomposition—disagreement would refute the spectral varianc","supporting_citations":[],"review_version":1}