{"id":"f8079c3d-f054-41b9-8b68-d95d55a82c75","arxiv_id":"1908.06735","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a Gaussian long-memory process sampled at Poisson times, the periodogram converges to a weighted chi-square limit with explicit d-dependent constants, and the local Whittle estimator of the memory parameter remains consistent.","lead":"This paper develops the spectral and estimation theory for Gaussian long-memory processes measured at random, Poisson-chosen times. The practical payoff: the memory parameter can still be estimated from irregularly spaced data, but confidence intervals need correction factors rather than the textbook chi-square.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3's second cumulant bound (30) is asserted without proof and the displayed argument only gives a weaker bound; Theorem 2's first-regime display depends on it.","rationale":"The reader's weakest assumption was Condition H_f, which is a legitimate hypothesis but not the most concrete correctness risk: H_f is an explicit, reasonably broad condition, and the paper's main theorems are conditional on it. The sharper issue is internal: Lemma 3's second inequality is stated, used in the proof of Theorem 2, and never proved. The proof that follows the lemma establishes only the sup-over-h bound and the (33) side bound; no argument for (30) appears. Summing the proved bound over h would give n^{1+2d}, which is larger than n^{4d} for d<1/2, so the displayed proof cannot justify (30). A direct cumulant estimate suggests the claimed n^{4d} rate may be wrong for small d, but the key point is that the proof as written relies on an unproved bound. This is load-bearing for the local-Whittle consistency theorem, though probably repairable: even the weaker n^{1+2d} bound would still yield Delta_m=o(m) for bandwidths n^a with a<1, so the theorem's conclusion is likely sound. The verdict should remain CONDITIONAL rather than being strengthened to acceptance, because the manuscript needs a completed proof of (30) or a corrected rate in the Theorem 2 argument.","tokens_in":16444,"tokens_out":39393,"duration_ms":386713,"concrete_test":"Compute S_n(d)=sum_{h,r,s=0}^n |cum(Y0,Yh,Yr,Ys)| for d=0.1 at n=10^3 and n=10^4, using the cumulant formula (34) with Gamma-moment covariances from Lemma 2 (or a high-accuracy Monte-Carlo version). Compare the growth to n^{4d}=n^{0.4}: if S_n scales like n rather than n^{0.4}, Lemma 3(30) fails as stated and Theorem 2's first-regime display must be amended or re-proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3 states two fourth-cumulant bounds for Y_n=X_{T_n}. The proof establishes the sup-over-h bound (29) and the auxiliary bound (33), then stops; the second bound (30), namely sum_{h,r,s=0}^n |cum(Y0,Yh,Yr,Ys)| <= C n^{4d}, is never proved. The natural route from (29), summing over h, yields only C n^{1+2d}, which is weaker (larger) than n^{4d} for every d<1/2. A direct estimate of the three covariance terms in (34) using Lemma 2 also suggests the true order may be n^{1+2d} for small d, so (30) is at least unsubstantiated and possibly false as stated. This matters because the first regime of Theorem 2's proof uses (30) verbatim to conclude Delta_m/m <= C(m^{-1/2}+m^{2d}/n^{1/2}); that step is therefore unsupported. The gap is likely patchable -- replacing n^{4d} by the cruder n^{1+2d} still gives Delta_m=o(m) for m=n^a, 0<a<1 -- but as written the local-Whittle consistency proof contains a missing proof at a load-bearing point.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the discrete-time process Y_n = X_{T_n} obtained by sampling a stationary continuous-time Gaussian long-memory process X at the arrival times of an independent Poisson process. It derives the spectral density of Y from that of X (Lemma 1), gives a local expansion of the sampled spectral density under exponential interarrivals (Proposition 1), and then uses this to prove three main results: the convergence of normalized periodogram ordinates to a d-dependent weighted chi-square-type vector (Theorem 1), the consistency of a local Whittle estimator at rate o_P(1/log n) (Theorem 2), and the consistency of a long-run variance estimator (Theorem 3).","tokens_in":16613,"tokens_out":18360,"duration_ms":176123,"significance":"If the main results hold, the paper provides a nontrivial extension of classical periodogram asymptotics to randomly sampled continuous-time long-memory processes. The explicit correction constants L_j(d), R_j(d) in Theorem 1 are new and potentially useful for inference, and the claim that local Whittle estimation remains consistent despite the loss of joint Gaussianity is valuable. The derivations are largely visible: Eq. (2) follows from a standard Abel-summation argument, Corollary 1 uses the exact Gamma moments of Poisson sampling in a transparent way, and the proof of the first cumulant bound in Lemma 3 is sufficiently detailed. However, the paper currently contains a load-bearing missing proof in Lemma 3 and relies on an omitted proof in Remark 1, so the published version needs additional technical work.","major_comments":[{"comment":"The proof of Lemma 3 establishes (29) and (33), then stops; the second bound (30) is asserted without proof. Summing (29) over h gives only O(n^{1+2d}), which is not O(n^{4d}) for d<1/2, and the natural estimate of the three covariance terms in (34) using Corollary 1 also gives O(n^{1+2d}) unless additional cancellations are proved. This matters because the first regime of the proof of Theorem 2 uses (30) verbatim to obtain in_{2}(k) ≤ C n^{-2}(m/n)^{4d} m^2 n n^{4d} and hence Δ_m/m ≤ C(m^{-1/2}+m^{2d}/n^{1/2}); without (30) that line is unsupported. The gap is likely patchable, but as written the local-Whittle consistency proof is incomplete.","section":"Section 4, Lemma 3 and Theorem 2"},{"comment":"Remark 1 extends Proposition 1 to Condition H_f with c(d) in place of φ(0) but omits the proof. The current proof of Proposition 1 relies on the local representation f_X(λ)=λ^{-2d}φ(λ) with φ differentiable at 0 and φ(0)≠0; under H_f the remainder is encoded by a monotone function h and the covariance expansion of Lemma 2, so the two-integral local analysis in Proposition 1 must be reworked. The extension is used by Theorem 1 (through the normalization by f_Y) and by Theorem 2 (through the form f_Y(λ)∼c|λ|^{-2d}). Since this is load-bearing, the omitted proof should be included or the statement should be proved as a separate lemma.","section":"Section 2, Remark 1"}],"minor_comments":[{"comment":"The sentence 'with Z1(j), Z2(k) are independent for all j, k' is easy to misread as saying that all entries of the limiting Gaussian vector are independent. Since (19)-(20) give nonzero within-family cross-frequency covariances, please rephrase as 'Z1(j) and Z2(k) are independent for every j,k' and explicitly note that correlations within the Z1 family and within the Z2 family are given by (19)-(20).","section":"Section 3, Theorem 1, Eqs. (17)-(20)"},{"comment":"The phrase 'using Lemma 1' at the end of display (28) should be 'using Corollary 1' (or Lemma 2), because the displayed bound uses Var(σ_X(T_h))=O(h^{-2+2d}).","section":"Section 3, proof of Theorem 1, display (28)"},{"comment":"The normalized periodogram η_j^* = I_n(λ_j)/(b λ_j^{-2d}) uses an unspecified constant b, while v(λ_j) later uses c. Please define b explicitly and coordinate the notation.","section":"Section 4.1, proof of Theorem 2"},{"comment":"The displayed formula for S_Y^2 contains n in the integrand through sin(nλ/2); after the limit in n is taken, the long-run variance should be n-free. Please correct the formula by an explicit change of variable or by writing the limiting constant directly.","section":"Section 4.2, Theorem 3"},{"comment":"Remark 2 refers to 'Proposition 2', but no Proposition 2 appears in the paper; the reference should be to Theorem 2.","section":"Section 4.2, Remark 2"},{"comment":"There are several typographical issues: the title reads 'randomly sample d processes' instead of 'randomly sampled processes', Section 1 contains 'joint-Gaussienty', the proof of Proposition 1 has 'using using', and the author affiliation contains 'F rance'. These should be corrected in a final pass.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript's central claims are plausible and the derivations are mostly transparent. The blocking issues are the unproved bound (30) in Lemma 3, on which Theorem 2's first regime depends, and the omitted proof of Remark 1, which underpins Theorems 1 and 2. I see no grounds for concern about novelty or attribution; the missing work is technical and within the paper's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my read before you sink time into it. This is a real contribution to the random-sampling long-memory literature, not a repackaging. Its new content is concrete: the full proof of the spectral density representation (2) under random sampling, the sharp low-frequency expansion f_Y(x)=x^{-2d}(phi(0)+sigma_X(0)(2pi)^{-1}x^{2d}+o(x^{2d})) for Poisson sampling, the weighted chi-square periodogram limit in Theorem 1 with explicit d-dependent constants, and the local Whittle consistency theorem. The fact that the sampled process is marginally Gaussian but not jointly Gaussian is exactly what makes Theorem 1 nontrivial, and the paper does not wave it away. The derivations are unusually visible; I checked the main rates—Corollary 1's Gamma moments, Lemma 3's first cumulant bound, the mean normalization in Theorem 1—and they line up. The citation pattern is reasonable; the self-citations are to the earlier results this paper extends, not padding.\n\nThe soft spot, and it is a real one, is Lemma 3. The proof establishes (29) and (33), then stops. The second stated bound, (30) with n^{4d}, is what Theorem 2 uses in the low-bandwidth regime, but the displayed argument never proves it, and summing (29) over h only gives n^{1+2d}, which is weaker than n^{4d} for d<1/2. To be fair, this is patchable: the high-bandwidth regime goes through a different argument, and a weaker bound still gives the desired Delta_m=o(m) for bandwidths up to n^{1/2}. But as written, the local-Whittle proof has a missing step at a load-bearing point.\n\nTwo smaller issues. Remark 1 says Proposition 1 carries over to Condition H_f with the proof omitted; that extension is exactly what Theorem 2 needs, so it should be written out or split into a proper lemma. And the paper invokes external results—Hurvich-Beltrao, Dalla et al., Abadir et al.—without checking their hypotheses in detail; probably fine, but a referee should verify. No simulations, which is fine for this genre, though they would help confirm the new periodogram constants.\n\nBottom line: this is a serious paper for the long-memory/random-sampling niche. The main ideas and theorem statements are credible, and the deficiencies are repairable. I would send it to a competent referee rather than desk-reject, with a specific request to nail down (30) and re-run the local-Whittle argument.","headline":"A dense but genuine contribution to inference for Poisson-sampled continuous-time long-memory processes; the weighted chi-square limit and local Whittle result are new, but Lemma 3's second cumulant bound is unproved as written and needs repair before the paper is fully acceptable.","tokens_in":17251,"tokens_out":5698,"would_cite":true,"duration_ms":53279,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62M15","62M10","60G15","60G55","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a Gaussian long-memory process sampled at Poisson times, periodogram ordinates converge to a d-dependent weighted chi-square, not the usual chi-square.","keywords":["long memory","random sampling","Poisson process","periodogram","local Whittle estimator","continuous-time Gaussian processes","spectral density","limit theorems"],"falsifier":"Simulate a stationary Gaussian process with spectral density $f_X(\\lambda)=|\\lambda|^{-2d}e^{-|\\lambda|}$, sample it at the arrival times of a unit-rate Poisson process, and compare the empirical distribution of the normalized periodogram ordinates $I_n(\\lambda_j)/f_Y(\\lambda_j)$ at fixed $j$ against Theorem 1's law. If the sample variances of the limiting variables deviate from $1/2 \\mp R_j(d)/L_j(d)$ beyond Monte Carlo error, or if the cross-frequency covariances do not match equations (19)--(20), the theorem is refuted.","tokens_in":16130,"feed_emoji":"🎲","tokens_out":8941,"duration_ms":77104,"temperature":0.7,"pith_summary":"The paper studies a continuous-time long-memory Gaussian process observed at random Poisson-distributed times, a sampling scheme that turns the data into a stationary but no longer jointly Gaussian discrete process. It shows that this sampled process keeps the same long-memory parameter at frequency zero, and proves an explicit limit theorem for the low-frequency periodogram: each normalized ordinate converges to $L_j(d)(Z_1(j)^2+Z_2(j)^2)$, with the $Z$'s Gaussian and with variances $1/2 \\mp R_j(d)/L_j(d)$ that depend on the memory parameter $d$ and the frequency $j$. This replaces the textbook chi-square limit for periodogram ordinates with a d-dependent correction, which matters because standard spectral inference would be systematically wrong for randomly sampled long-memory data. The paper also proves that the local Whittle estimator of $d$ remains consistent at the rate $o_P(1/\\log n)$, and that the long-run variance can be estimated consistently.","feed_headline":"Random sampling changes the periodogram limit law","feed_subtitle":"For long-memory Gaussian processes, the usual chi-square limit becomes a weighted chi-square that depends on the memory parameter d.","key_machinery":"The engine of the paper is the covariance decomposition of Lemma 2: under Condition $H_f$, $\\sigma_X(x)=c(d)x^{2d-1}+g(x)$ with $|g(x)|\\le C(d)x^{-1}$. This single bound, integrated against the exact Gamma moments of Poisson arrival times, produces Corollary 1 and the fourth-cumulant estimates of Lemma 3. These cumulant bounds are what allow the conditional-Gaussian characteristic-function argument in Theorem 1 and the local Whittle proof in Theorem 2 to go through for a process that is not jointly Gaussian and not linear. The spectral identity $f_Y(x)=x^{-2d}f_Y^*(x)$ with $f_Y^*(x)=\\varphi(0)+\\sigma_X(0)x^{2d}/(2\\pi)+o(x^{2d})$ is the frequency-domain counterpart that connects the sampled process to standard long-memory asymptotics.","core_discovery":"Under Condition $H_f$, the spectral density of $X$ factors as $c|\\lambda|^{-2d}(1-h(\\lambda))$; when $X$ is sampled at the arrival times of a unit-rate Poisson process, the discrete process $Y_n=X_{T_n}$ has spectral density $f_Y(x)=x^{-2d}f_Y^*(x)$ with $f_Y^*(x)=\\varphi(0)+\\sigma_X(0)x^{2d}/(2\\pi)+o(x^{2d})$. The paper's main theorem shows that for any fixed set of distinct low Fourier frequencies $\\lambda_j=2\\pi j/n$, the vector of normalized periodogram ordinates $(I_n(\\lambda_j)/f_Y(\\lambda_j))$ converges in distribution to the vector $L_j(d)(Z_1^2(j)+Z_2^2(j))$, with the $Z$'s Gaussian, variances $1/2 \\mp R_j(d)/L_j(d)$, and cross-frequency covariances given by equations (19)--(20). Because $R_j(d)$ is generally nonzero, the classical $\\chi^2_2$ asymptotic behavior does not survive random sampling; instead, the limit is a weighted chi-square-type law depending explicitly on $d$ and on the frequency $j$. The proof works by conditioning on the Poisson arrival times, where the sample is jointly Gaussian, then showing that the conditional covariance matrix converges to an explicit limit; the variance of the conditional covariance matrix vanishes via the bounds $\\operatorname{Var}(\\sigma_X(T_r))=O(r^{-2+2d})$ derived from Lemma 2. A further set of fourth-cumulant bounds yields consistency of the local Whittle estimator at the $o_P(1/\\log n)$ rate.","pith_inferences":["An immediate numerical test of Theorem 1 would simulate a Gaussian process with a spectrum satisfying $H_f$, sample at exponential times, and compare empirical quantiles of $I_n(\\lambda_j)/f_Y(\\lambda_j)$ with the stated weighted chi-square law; this would also reveal how quickly the asymptotic correction kicks in.","The constants $L_j(d)$ and $R_j(d)$ suggest a practical correction factor for spectral density estimates of randomly sampled data; correcting by these factors before estimating $d$ could reduce the bias that the usual periodogram-based methods would have.","If the sampling intervals have a distribution other than exponential, the Gamma-moment calculations would change; the method of proof suggests that analogous results hold whenever the renewal process has moments making $\\operatorname{Var}(\\sigma_X(T_r))=O(r^{-2+2d})$, but that extension is not proved here.","The local Whittle rate $o_P(1/\\log n)$ is slower than the parametric $\\sqrt{n}$ rate; for long-memory data this is typical, but the paper leaves open whether a sharper rate can be obtained for the non-Gaussian sampled process under stronger conditions."],"forward_implications":["Any inference procedure for Poisson-sampled continuous-time long-memory data that assumes the standard $\\chi^2_2$ limit for periodogram ordinates is asymptotically miscalibrated; Theorem 1 supplies the corrected weighted chi-square limit with explicit constants.","The local Whittle estimator remains consistent for the memory parameter $d$ even though $Y_n$ is neither linear nor jointly Gaussian, and with bandwidth $m_n=n^a$ it reaches $o_P(1/\\log n)$.","The long-run variance of the sampled process can be estimated consistently from the sample covariances with Bartlett weights, enabling mean and stationarity testing for such data.","Poisson sampling does not erase long memory: the sampled spectral density behaves as $c|\\lambda|^{-2d}$ near zero, so the memory parameter can still be read off from low-frequency behavior.","The fourth-order cumulant bounds of Lemma 3 hold for the sampled process, making it tractable for limit theorems despite heavy dependence."],"supporting_citations":[{"why":"It supplies the low-frequency periodogram covariance asymptotics used to identify the limit of the conditional covariance matrix in Theorem 1.","marker":"Hurvich and Beltrão (1993)"},{"why":"It provides the cumulant-based framework and lemmas that Theorem 2 uses to prove local Whittle consistency.","marker":"Dalla et al. (2006)"},{"why":"Its long-run variance estimator theorem is invoked to prove Theorem 3 once the fourth-cumulant condition is verified.","marker":"Abadir et al. (2009)"},{"why":"It established that random sampling of a Gaussian process destroys joint Gaussianity, the phenomenon this paper builds on.","marker":"Philippe et al. (2018)"},{"why":"It showed that random sampling can preserve or reduce long memory, the background for the spectral-density behavior studied here.","marker":"Philippe and Viano (2010)"},{"why":"It supplies the integral identities used in Lemma 2 and Proposition 1 for the covariance and spectral expansions.","marker":"Gradshteyn and Ryzhik (2015)"}],"fun_headline_variants":["Random sampling turns periodogram limit into weighted chi-square","Long memory periodogram limit depends on d under Poisson sampling","Poisson sampling: periodogram limit law becomes d-dependent","Random sampling breaks chi-square periodogram limit for long memory","Weighted chi-square periodogram limit under random sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything downstream of Lemma 2 depends on Condition $H_f$, which requires the spectral density of $X$ to factor as $c|\\lambda|^{-2d}(1-h(\\lambda))$ with $h(\\lambda)$ nondecreasing, $h(0)=0$, $h\\to 1$, and differentiable at 0; if a spectrum does not admit this monotone-remainder split, the covariance bound that feeds every variance and cumulant estimate in the paper is not established.","fun_headline_variants_meta":{"raw":{"variants":["Random sampling turns periodogram limit into weighted chi-square","Long memory periodogram limit depends on d under Poisson sampling","Poisson sampling: periodogram limit law becomes d-dependent","Random sampling breaks chi-square periodogram limit for long memory","Weighted chi-square periodogram limit under random sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000739,"raw_usage":{"total_tokens":3299,"prompt_tokens":946,"completion_tokens":2353,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":2274}},"tokens_in":562,"tokens_out":2353,"duration_ms":17319,"temperature":1.0,"reasoning_tokens":2274,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:41:46.303898+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a stationary Gaussian process with spectral density $f_X(\\lambda)=|\\lambda|^{-2d}e^{-|\\lambda|}$, sample it at the arrival times of a unit-rate Poisson process, and compare the empirical distribution of the normalized periodogram ordinates $I_n(\\lambda_j)/f_Y(\\lambda_j)$ at fixed $j$ against Theorem 1's law. If the sample variances of the limiting variables deviate from $1/2 \\mp R_j(d)/L_j(d)$ beyond Monte Carlo error, or if the cross-frequency covariances do not match equations (19)--(20), the theorem is refuted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the low-frequency periodogram covariance asymptotics used to identify the limit of the conditional covariance matrix in Theorem 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the cumulant-based framework and lemmas that Theorem 2 uses to prove local Whittle consistency."},{"cited_title":"M., Distaso, W., and Giraitis, L","cited_arxiv_id":null,"evidence_quote":"Its long-run variance estimator theorem is invoked to prove Theorem 3 once the fourth-cumulant condition is verified."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It established that random sampling of a Gaussian process destroys joint Gaussianity, the phenomenon this paper builds on."},{"cited_title":"and Viano, M.-C","cited_arxiv_id":null,"evidence_quote":"It showed that random sampling can preserve or reduce long memory, the background for the spectral-density behavior studied here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the integral identities used in Lemma 2 and Proposition 1 for the covariance and spectral expansions."}],"review_version":1}