Inference for continuous-time long memory randomly sampled processes

Anne Philippe (LMJL); Caroline Robet (LMJL); Mohamedou Ould Haye

REVIEW 2 major objections 6 minor 18 references

Inference for continuous-time long memory randomly sampled processes

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.06735 v2 pith:TZDXYJYZ submitted 2019-08-19 math.ST stat.TH

Mohamedou Ould Haye , Anne Philippe (LMJL) , Caroline Robet (LMJL) This is my paper · ORCID

classification math.STstat.TH MSC 62M1562M1060G1560G5560F05

keywords longmemoryrandomsamplingPoissonprocessperiodogramlocalWhittleestimatorcontinuous-timeGaussianprocessesspectraldensitylimittheorems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

Two 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.

Bottom 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.

Referee Report

2 major / 6 minor

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).

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 (2)

[Section 4, Lemma 3 and Theorem 2] 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 2, Remark 1] 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.

minor comments (6)

[Section 3, Theorem 1, Eqs. (17)-(20)] 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 3, proof of Theorem 1, display (28)] 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 4.1, proof of Theorem 2] 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 4.2, Theorem 3] 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 4.2, Remark 2] Remark 2 refers to 'Proposition 2', but no Proposition 2 appears in the paper; the reference should be to Theorem 2.
[Throughout] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: limits are computed from assumptions and external theorems; the only self-citations are peripheral, and a separate omitted proof in Lemma 3 is a correctness gap, not circularity.

full rationale

The paper's central claims do not reduce to their inputs by construction. Theorem 1 is derived by conditioning on the Poisson sampling times, where the discrete Fourier transforms are exactly Gaussian; the conditional covariance is written in terms of sigma_X, E(Sigma_T)->Sigma is taken from the external Hurvich and Beltrao (1993) result, and Var(Sigma_T)->0 is driven by Lemma 2 and Corollary 1, which are proven from Condition H_f via integration by parts and exact Gamma moments. The constants L_j(d), R_j(d), L_{j,k}(d), R_{j,k}(d) are explicit integrals depending on the spectral singularity, not fitted parameters renamed as predictions. Theorem 2 follows the standard local-Whittle route of Dalla et al. (2006) with the cumulant bounds of Lemma 3; again no estimate is recycled as a predicted quantity. The self-citations to Philippe et al. (2018) (loss of joint Gaussianity and sample-mean CLT) and Philippe and Viano (2010) (memory preservation under random sampling) are used as motivation and context rather than as the engine of Theorems 1-3; no uniqueness or alternative-exclusion claim is imported from the authors' own work. The genuine problem found in the paper is a non-circular gap: Lemma 3's second bound, equation (30), is asserted after the proof of (33) but never proved; the displayed estimates establish (29) and (33), and Theorem 2's first regime uses (30) verbatim to conclude Delta_m/m <= C(m^{-1/2}+m^{2d}/n^{1/2}). Because this is an omitted proof rather than a definitional or fitted equivalence, it does not raise the circularity score, though it is a load-bearing correctness risk.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data; there is no data. The estimators involve user-chosen bandwidths m_n = n^a (Theorem 2) and q = o(sqrt(n)) (Theorem 3), which are standard tuning choices, not fitted constants. The constants c(d), C(d), L_j(d), R_j(d) are derived from the spectral model, not fitted. No new physical or mathematical entities are postulated: the limit variables Z_1(j), Z_2(j) are derived from the spectral density, not introduced as independent objects. The axioms list the model assumptions (Gaussian X, H_f, Poisson sampling), the imported external theorems, the unproved Remark 1, and the self-cited CLT.

assumptions (6)

domain assumption X is a zero-mean, strictly stationary continuous-time Gaussian process with spectral density f_X(lambda) = c|lambda|^(-2d)(1 - h(lambda)), 0 < d < 1/2, satisfying Condition H_f.
Stated in Sections 1 and 2 and used throughout; all theorems are conditional on this model.
domain assumption Sampling times T_n form a rate-1 Poisson process (i.i.d. exponential intervals) independent of X.
Corollary 1 and Lemma 3 rely on exact Gamma moments of T_r; Theorems 1 through 3 are proved only for this case.
standard math External asymptotics valid: Theorem 5 of Hurvich and Beltao (1993), Proposition 3 and Lemma 4 of Dalla et al. (2006), Theorem 2.2 of Abadir et al. (2009).
Used as black boxes in Theorem 1 (mean of periodogram), Theorem 2 (local Whittle consistency and variance bound), and Theorem 3 (long-run variance); their hypotheses are stated only loosely in the text.
ad hoc to paper Remark 1: under H_f, Proposition 1 holds with c(d) in place of phi(0); proof is omitted.
Unproved bridge used by Theorem 2 to obtain f_Y(x) ~ c x^(-2d).
domain assumption Sample-mean CLT n^(1/2-d)(Ybar_n - mu) converges to N(0, .) from Philippe et al. (2018).
Section 4.2 motivates long-run variance estimation using this CLT; the cited paper is a working preprint by the same group.
standard math Integral formulas 2.556.1 and 3.761.9 of Gradshteyn and Ryzhik (2015) for the arctan integral and the cosine transform of lambda^(-2d).
Used in Lemma 1 (the V function) and Lemma 2 (cosine transform computation).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Inference for continuous-time long memory randomly sampled processes." pith.science (2026). https://pith.science/paper/TZDXYJYZ

@misc{pith2026190806735,
  author       = {Pith},
  title        = {Pith review of: Inference for continuous-time long memory randomly sampled processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZDXYJYZ}},
  note         = {Machine review of arXiv:1908.06735}
}

read the original abstract

From a continuous-time long memory stochastic process, a discrete-time randomly sampled one is drawn. We investigate the second-order properties of this process and establish some time-and frequency-domain asymptotic results. We mainly focus on the case when the initial process is Gaussian. The challenge being that, although marginally remains Gaussian, the randomly sampled process will no longer be jointly Gaussian.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

M., Distaso, W., and Giraitis, L

Abadir, K. M., Distaso, W., and Giraitis, L. (2009). Two estimators of the long-run variance: beyond short memory. J. Econometrics , 150(1):56--70

work page 2009
[2]

and Bertrand, P

Bardet, J.-M. and Bertrand, P. (2010). A non-parametric estimator of the spectral density of a continuous-time gaussian process observed at random times. Scandinavian Journal of Statistics , 37:458--476

work page 2010
[3]

Beran, J., Feng, Y., Ghosh, S., and Kulik, R. (2013). Long-memory processes . Springer, Heidelberg. Probabilistic properties and statistical methods

work page 2013
[4]

Chambers, M. J. (1996). The estimation of continuous parameter long-memory time series models. Econometric Theory , 12(2):374--390

work page 1996
[5]

Comte, F. (1996). Simulation and estimation of long memory continuous time models. J. Time Ser. Anal. , 17(1):19--36

work page 1996
[6]

and Renault, E

Comte, F. and Renault, E. (1996). Long memory continuous time models. J. Econometrics , 73(1):101--149

work page 1996
[7]

Dalla, V., Giraitis, L., and Hidalgo, J. (2006). Consistent estimation of the memory parameter for nonlinear time series. J. Time Ser. Anal. , 27(2):211--251

work page 2006
[8]

deLyra, J. L. (2015). Fourier theory on the complex plane IV : Representability of real functions by their fourier coefficients. working paper or preprint, arXiv:1502.01617

work page Pith review arXiv 2015

Show all 18 references

[9]

Doukhan, P., Oppenheim, G., and Taqqu, M. (2003). Theory and Applications of Long-Range Dependence . Birkh\"auser, Boston

2003
[10]

L., and Surgailis, D

Giraitis, L., Koul, H. L., and Surgailis, D. (2012). Large S ample I nference for L ong M emory P rocesses . Imperial College Press, London

2012
[11]

Giraitis, L., Leipus., R., and Philippe, A. (2006). A test for stationarity versus trends and unit roots for a wide class of dependent errors. Econometric Theory , 22:989--1029

2006
[12]

Gradshteyn, I. S. and Ryzhik, I. M. (2015). Table of integrals, series, and products . Elsevier/Academic Press, Amsterdam, eighth edition. Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition ...

2015
[13]

Hurvich, C. M. and Beltr\ a o, K. I. (1993). Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time series. J. Time Ser. Anal. , 14(5):455--472

1993
[14]

Philippe, A., Robet, C., and Viano, M.-C. (2018). Random discretization of stationary continuous time processes . working paper or preprint

2018
[15]

and Viano, M.-C

Philippe, A. and Viano, M.-C. (2010). Random sampling of long-memory stationary processes. Journal of Statistical Planning and Inference , 140(5):1110 -- 1124

2010
[16]

and Chan, K

Tsai, H. and Chan, K. S. (2005a). Maximum likelihood estimation of linear continuous time long memory processes with discrete time data. J. R. Stat. Soc. Ser. B Stat. Methodol. , 67(5):703--716

2005
[17]

and Chan, K

Tsai, H. and Chan, K. S. (2005b). Quasi-maximum likelihood estimation for a class of continuous-time long-memory processes. J. Time Ser. Anal. , 26(5):691--713

2005
[18]

Viano, M.-C., Deniau, C., and Oppenheim, G. (1994). Continuous-time fractional ARMA processes. Statist. Probab. Lett. , 21(4):323--336

1994

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

M., Distaso, W., and Giraitis, L

Abadir, K. M., Distaso, W., and Giraitis, L. (2009). Two estimators of the long-run variance: beyond short memory. J. Econometrics , 150(1):56--70

work page 2009

[2] [2]

and Bertrand, P

Bardet, J.-M. and Bertrand, P. (2010). A non-parametric estimator of the spectral density of a continuous-time gaussian process observed at random times. Scandinavian Journal of Statistics , 37:458--476

work page 2010

[3] [3]

Beran, J., Feng, Y., Ghosh, S., and Kulik, R. (2013). Long-memory processes . Springer, Heidelberg. Probabilistic properties and statistical methods

work page 2013

[4] [4]

Chambers, M. J. (1996). The estimation of continuous parameter long-memory time series models. Econometric Theory , 12(2):374--390

work page 1996

[5] [5]

Comte, F. (1996). Simulation and estimation of long memory continuous time models. J. Time Ser. Anal. , 17(1):19--36

work page 1996

[6] [6]

and Renault, E

Comte, F. and Renault, E. (1996). Long memory continuous time models. J. Econometrics , 73(1):101--149

work page 1996

[7] [7]

Dalla, V., Giraitis, L., and Hidalgo, J. (2006). Consistent estimation of the memory parameter for nonlinear time series. J. Time Ser. Anal. , 27(2):211--251

work page 2006

[8] [8]

deLyra, J. L. (2015). Fourier theory on the complex plane IV : Representability of real functions by their fourier coefficients. working paper or preprint, arXiv:1502.01617

work page Pith review arXiv 2015