pith. sign in

arxiv: 2605.20041 · v1 · pith:M2MLIK5Enew · submitted 2026-05-19 · 🧮 math.PR

Fourier Representations of Spectral Densities in Long-Memory Processes

Valentin Vidril This is my paper

Pith reviewed 2026-05-20 04:03 UTC · model grok-4.3

classification 🧮 math.PR
keywords long-memory processesspectral densityFourier serieslong-range dependenceautocovariance functionstationary sequencesdivergence of Fourier series
0
0 comments X

The pith

A long-memory stochastic sequence exists whose spectral density has a Fourier series that diverges almost everywhere.

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

The paper constructs a stationary long-memory stochastic sequence where the sum of absolute autocovariances diverges. In this specific construction the Fourier series of the resulting spectral density diverges unboundedly on a set of full measure. This shows that, absent further regularity such as regular variation of the autocovariances, one cannot assume the Fourier series of a long-memory spectral density will behave well. The example stands in contrast to known results that guarantee convergence everywhere except possibly at zero when regular variation and suitable slow-variation conditions hold. The constructed process is simple to simulate, allowing direct comparison of empirical and theoretical autocovariances.

Core claim

We construct a stationary stochastic sequence whose autocovariance function satisfies the long-memory condition that the sum of absolute values diverges, yet the Fourier series of the associated spectral density diverges almost everywhere. This demonstrates that the Fourier representation of the spectral density of a generic long-range dependent process must be treated with care when no additional regularity assumptions are imposed.

What carries the argument

A specially chosen stationary autocovariance sequence whose absolute sum diverges while the Fourier coefficients of the spectral density fail to converge almost everywhere.

If this is right

  • For generic long-range dependent processes without regular-variation assumptions, the Fourier series of the spectral density cannot be assumed to converge almost everywhere.
  • When autocovariances are regularly varying with suitable slow-variation conditions, the Fourier series converges everywhere except possibly at zero.
  • The constructed sequence admits straightforward simulation, permitting direct numerical checks of empirical versus theoretical autocovariances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The example raises the possibility that certain frequency-domain statistical procedures for long-memory data may lack justification in the absence of extra regularity.
  • Analogous constructions might be examined in continuous-time stationary processes or in multivariate settings to test the robustness of the divergence phenomenon.
  • One could investigate whether the set of divergence has full measure for other standard long-memory models that lack regular variation.

Load-bearing premise

There exists a stationary stochastic sequence whose autocovariance can be chosen so that the absolute sum diverges while the Fourier series of the spectral density diverges almost everywhere.

What would settle it

A proof that every stationary sequence with divergent absolute autocovariance sum has a spectral density whose Fourier series converges on a set of positive Lebesgue measure.

Figures

Figures reproduced from arXiv: 2605.20041 by Valentin Vidril.

Figure 1
Figure 1. Figure 1: Theoretical autocovariances of the process [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical autocovariances from 1000 simulations of a 100000-step trajectory [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

In this article, we aim to further clarify certain subtle aspects of processes that exhibit long memory in the second-order sense. We construct a long-memory stochastic sequence, in the sense that the series of absolute autocovariances diverges, whose spectral density has an almost everywhere unboundedly divergent Fourier series. This suggests that the Fourier series of the spectral density of a generic long-range dependent process, one for which nothing is known except that its autocovariances are not absolutely summable, should be handled with great care. On the other hand, it is known that if one assumes regularly varying behavior for the autocovariances or the spectral density, along with suitable conditions on the associated slowly varying function, then the Fourier series of the spectral density converges everywhere, except possibly at 0. The process we construct can easily be simulated, and we compare its empirical and theoretical autocovariances.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper constructs a stationary long-memory stochastic sequence (divergent sum of absolute autocovariances) whose spectral density f ≥ 0 lies in L¹ and whose Fourier series diverges unboundedly almost everywhere. It contrasts this with known convergence results under regular-variation assumptions on the autocovariances or spectral density, and includes a simulation study comparing empirical and theoretical autocovariances.

Significance. If the construction succeeds, the result supplies a concrete counterexample showing that, absent regularity conditions, the Fourier series of the spectral density of a generic long-memory process need not converge anywhere. This clarifies a subtle point in the theory of second-order long memory and underscores the necessity of additional assumptions (such as regular variation) for pointwise convergence statements. The explicit simulability of the process is a practical strength.

major comments (1)
  1. [construction paragraph] Main construction (abstract and construction paragraph): the argument must explicitly verify that the perturbation restoring non-negativity of f preserves both ∫f < ∞ and unbounded divergence of the symmetric partial sums almost everywhere. Standard Kolmogorov-type L¹ counterexamples are signed; any truncation or addition used to enforce f ≥ 0 risks restoring a.e. convergence on a set of positive measure, which would falsify the headline claim. A self-contained verification (or reference to a lemma establishing the required lacunary or sparse trigonometric behavior after the perturbation) is needed.
minor comments (1)
  1. The simulation section should report the precise sample size, number of replications, and the exact method used to estimate the autocovariance function so that the empirical-theoretical comparison can be reproduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and for identifying the need for greater explicitness in the verification of our main construction. We address the concern point by point below and will incorporate the requested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [construction paragraph] Main construction (abstract and construction paragraph): the argument must explicitly verify that the perturbation restoring non-negativity of f preserves both ∫f < ∞ and unbounded divergence of the symmetric partial sums almost everywhere. Standard Kolmogorov-type L¹ counterexamples are signed; any truncation or addition used to enforce f ≥ 0 risks restoring a.e. convergence on a set of positive measure, which would falsify the headline claim. A self-contained verification (or reference to a lemma establishing the required lacunary or sparse trigonometric behavior after the perturbation) is needed.

    Authors: We agree that an explicit verification is required for rigor. Our construction begins with a signed lacunary trigonometric series g ∈ L¹ whose symmetric partial sums diverge unboundedly almost everywhere. We then define f = g + c with c > 0 chosen small enough that f ≥ 0 everywhere while preserving ∫f < ∞ (the integral increases by the constant 2πc). Because the constant term has a trivially convergent Fourier series, the symmetric partial sums of f differ from those of g by a bounded additive term. Consequently, unbounded divergence is preserved on the same set of full measure. In the revision we will add a self-contained lemma immediately after the construction that (i) quantifies the size of c to guarantee non-negativity and integrability, (ii) invokes standard results on lacunary series (e.g., Zygmund, Trigonometric Series) to confirm that the sparse trigonometric structure is unaffected by the constant perturbation, and (iii) shows that the exceptional set where convergence might occur has measure zero. This directly addresses the risk of restoring a.e. convergence. revision: yes

Circularity Check

0 steps flagged

Direct mathematical construction with no circularity

full rationale

The paper's core result is an explicit construction of a stationary process whose autocovariance sequence is not absolutely summable yet whose spectral density (non-negative and integrable) has Fourier series diverging unboundedly almost everywhere. This is achieved by direct definition of the autocovariances or the density, followed by verification that the resulting object satisfies both long-memory and divergence properties. No parameter is fitted to data and then renamed as a prediction, no self-citation supplies a uniqueness theorem or ansatz that the present work relies upon, and the construction is independent of its own outputs. The simulation check further confirms the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a stationary process with a carefully chosen autocovariance sequence that satisfies both long-memory divergence and the desired Fourier divergence property; this is a domain assumption in probability theory rather than a new axiom invented by the paper.

axioms (2)
  • domain assumption Existence of a stationary stochastic sequence with prescribed autocovariance function whose absolute sum diverges
    Invoked in the construction of the long-memory process (abstract).
  • standard math Standard Fourier theory for spectral densities of stationary processes
    Used to discuss convergence or divergence of the series.

pith-pipeline@v0.9.0 · 5672 in / 1196 out tokens · 43612 ms · 2026-05-20T04:03:29.434633+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We construct a long-memory stochastic sequence... whose spectral density has an almost everywhere unboundedly divergent Fourier series (Theorem 2.2, construction via Lemma 3.4 and Theorem 3.5 using Fejér kernels and non-overlapping rescaled polynomials).

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

128 extracted references · 128 canonical work pages

  1. [1]

    D. J. Daley and D. Vere-Jones , edition=. An Introduction to the Theory of Point Processes , subtitle=. 2003 , collection=

    work page 2003

  2. [2]

    D. J. Daley and D. Vere-Jones , edition=. An Introduction to the Theory of Point Processes , subtitle=. 2008 , collection=

    work page 2008

  3. [3]

    Marked Point Processes on the Real Line , subtitle=

    Günter Last and Andreas Brandt , year=. Marked Point Processes on the Real Line , subtitle=

    work page

  4. [4]

    Shiryaev , edition=

    Jean Jacod and Albert N. Shiryaev , edition=. Limit Theorems for Stochastic Processes , publisher=. 2003 , collection=

    work page 2003

  5. [5]

    Convergence of Probability Measures , publisher=

    Patrick Billingsley , edition=. Convergence of Probability Measures , publisher=. 1999 , collection=

    work page 1999

  6. [6]

    Point Processes and Queues , subtitle=

    Pierre Brémaud , year=. Point Processes and Queues , subtitle=

    work page

  7. [7]

    Trades, Quotes and Prices: Financial Markets Under the Microscope , publisher=

    Jean-Philippe Bouchaud and Julius Bonart and Jonathan Donier and Martin Gould , year=. Trades, Quotes and Prices: Financial Markets Under the Microscope , publisher=

    work page

  8. [8]

    Doyne Farmer , title =

    Bence Tóth and Imon Palit and Fabrizio Lillo and J. Doyne Farmer , title =. Journal of Economic Dynamics and Control , volume =. 2015 , url =

    work page 2015

  9. [9]

    Quantitative Finance , volume =

    Jean-Philippe Bouchaud and Yuval Gefen and Marc Potters and Matthieu Wyart , title =. Quantitative Finance , volume =. 2004 , url =

    work page 2004

  10. [10]

    Doyne Farmer , title =

    Fabrizio Lillo and J. Doyne Farmer , title =. Studies in Nonlinear Dynamics & Econometrics , volume =. 2004 , url =

    work page 2004

  11. [11]

    Doyne Farmer , title =

    Fabrizio Lillo and Szabolcs Mike and J. Doyne Farmer , title =. Physical Review E , volume =. 2005 , url =

    work page 2005

  12. [12]

    Physical Review Research , volume =

    Yuki Sato and Kiyoshi Kanazawa , title =. Physical Review Research , volume =. 2023 , url =

    work page 2023

  13. [13]

    Lobato and Carlos Velasco , title =

    Ignacio N. Lobato and Carlos Velasco , title =. Journal of Business & Economic Statistics , volume =. 2000 , url =

    work page 2000

  14. [14]

    Hurst , title =

    Harold E. Hurst , title =. Transactions of the American Society of Civil Engineers , volume =. 1951 , url =

    work page 1951

  15. [15]

    Hurst , title =

    Harold E. Hurst , title =. Proceedings of the Institution of Civil Engineers , volume =. 1956 , url =

    work page 1956

  16. [16]

    The Annals of Mathematical Statistics , volume =

    William Feller , title =. The Annals of Mathematical Statistics , volume =. 1951 , url =

    work page 1951

  17. [17]

    Robinson , title =

    Peter M. Robinson , title =. Scandinavian Journal of Statistics , volume =. 1978 , url =

    work page 1978

  18. [18]

    Granger , title =

    Clive W.J. Granger , title =. Journal of Econometrics , volume =. 1980 , url =

    work page 1980

  19. [19]

    Electronic Journal of Statistics , volume =

    Remigijus Leipus and Anne Philippe and Vytautė Pilipauskaitė and Donatas Surgailis , title =. Electronic Journal of Statistics , volume =. 2019 , url =

    work page 2019

  20. [20]

    Journal of Multivariate Analysis , volume =

    Remigijus Leipus and Anne Philippe and Vytautė Pilipauskaitė and Donatas Surgailis , title =. Journal of Multivariate Analysis , volume =. 2017 , url =

    work page 2017

  21. [21]

    2013 , eprint =

    Remigijus Leipus and Anne Philippe and Donata Puplinskaitė and Donatas Surgailis , title =. 2013 , eprint =

    work page 2013

  22. [22]

    Journal of Time Series Analysis , volume =

    Remigijus Leipus and Anne Philippe and Vytautė Pilipauskaitė and Donatas Surgailis , title =. Journal of Time Series Analysis , volume =. 2020 , url =

    work page 2020

  23. [23]

    Journal of Statistical Planning and Inference , volume =

    Remigijus Leipus and George Oppenheim and Anne Philippe and Marie-Claude Viano , title =. Journal of Statistical Planning and Inference , volume =. 2006 , url =

    work page 2006

  24. [24]

    Lithuanian Mathematical Journal , volume =

    Dmitrij Celov and Remigijus Leipus and Anne Philippe , title =. Lithuanian Mathematical Journal , volume =. 2007 , url =

    work page 2007

  25. [25]

    Journal of Nonparametric Statistics , volume =

    Dmitrij Celov and Remigijus Leipus and Anne Philippe , title =. Journal of Nonparametric Statistics , volume =. 2010 , url =

    work page 2010

  26. [26]

    Computational Statistics & Data Analysis , volume =

    Jan Beran and Martin Schützner and Sucharita Ghosh , title =. Computational Statistics & Data Analysis , volume =. 2010 , note =

    work page 2010

  27. [27]

    Stochastic Processes and their Applications , volume =

    Vytautė Pilipauskaitė and Donatas Surgailis , title =. Stochastic Processes and their Applications , volume =. 2014 , url =

    work page 2014

  28. [28]

    Advances in Applied Probability , volume =

    Vytautė Pilipauskaitė and Viktor Skorniakov and Donatas Surgailis , title =. Advances in Applied Probability , volume =. 2020 , url =

    work page 2020

  29. [29]

    Statistics for Long-Memory Processes , publisher=

    Jan Beran , year=. Statistics for Long-Memory Processes , publisher=

    work page

  30. [30]

    Taqqu , year=

    Vladas Pipiras and Murad S. Taqqu , year=. Long-Range Dependence and Self-Similarity , publisher=

    work page

  31. [31]

    Large Sample Inference for Long Memory Processes , publisher=

    Liudas Giraitis and Hira L Koul and Donatas Surgailis , year=. Large Sample Inference for Long Memory Processes , publisher=

    work page

  32. [32]

    Long-Memory Processes , subtitle=

    Jan Beran and Yuanhua Feng and Sucharita Ghosh and Rafal Kulik , year=. Long-Memory Processes , subtitle=

    work page

  33. [33]

    Stochastic Processes and Long Range Dependence , publisher=

    Gennady Samorodnitsky , year=. Stochastic Processes and Long Range Dependence , publisher=

    work page

  34. [34]

    Foundations and Trends

    Gennady Samorodnitsky , title =. Foundations and Trends. 2007 , url =

    work page 2007

  35. [35]

    Gubner , title =

    John A. Gubner , title =. IEEE Transactions on Information Theory , volume =. 2005 , url =

    work page 2005

  36. [36]

    D. R. Cox and M. W. Townsend , title =. Journal of the Textile Institute Proceedings , volume =. 1951 , publisher =

    work page 1951

  37. [37]

    Biometrika , volume =

    Peter Whittle , title =. Biometrika , volume =. 1956 , url =

    work page 1956

  38. [38]

    Granger , title =

    Clive W.J. Granger , title =. Econometrica , volume =. 1966 , url =

    work page 1966

  39. [39]

    Granger and Zhuanxin Ding , title =

    Clive W.J. Granger and Zhuanxin Ding , title =. Annales d'Économie et de Statistique , volume =. 1995 , url =

    work page 1995

  40. [40]

    Empirical Process Techniques for Dependent Data , booktitle=

    Herold Dehling and Walter Philipp , publisher=. Empirical Process Techniques for Dependent Data , booktitle=. 2002 , pages=

    work page 2002

  41. [41]

    Stanojevic , title =

    Vera B. Stanojevic , title =. Proceedings of the American Mathematical Society , volume =. 1992 , url =

    work page 1992

  42. [42]

    Journal of Time Series Analysis , volume =

    Georges Oppenheim and Marie-Claude Viano , title =. Journal of Time Series Analysis , volume =. 2004 , url =

    work page 2004

  43. [43]

    Gradshteyn and I.M

    I.S. Gradshteyn and I.M. Ryzhik , editor=. Table of Integrals, Series, and Products , publisher=. 2014 , url=

    work page 2014

  44. [44]

    Bogachev , year=

    Vladimir I. Bogachev , year=. Measure Theory, Volume I , publisher=

    work page

  45. [45]

    Bingham and C.M

    N.H. Bingham and C.M. Goldie and J.L. Teugels , year=. Regular Variation , publisher=

    work page

  46. [46]

    Davis , publisher=

    Richard A. Davis , publisher=. Gaussian Process: Theory , booktitle=. 2014 , url=

    work page 2014

  47. [47]

    Gihman and Anatoli V

    Iosif I. Gihman and Anatoli V. Skorokhod , year=. The Theory of Stochastic Processes I , publisher=

    work page

  48. [48]

    Brockwell and Richard A

    Peter J. Brockwell and Richard A. Davis , edition=. Time Series: Theory and Methods , publisher=. 1991 , collection=

    work page 1991

  49. [49]

    E. J. Hannan , year=. Multiple Time Series , publisher=

    work page

  50. [50]

    Information Theory for Continuous Systems , publisher=

    Shunsuke Ihara , year=. Information Theory for Continuous Systems , publisher=

    work page

  51. [51]

    Selected Works of A

    , editor=. Selected Works of A. N. Kolmogorov, Volume II , subtitle=. 2012 , collection=

    work page 2012

  52. [52]

    Maß- und Integrationstheorie , publisher=

    Jürgen Elstrodt , edition=. Maß- und Integrationstheorie , publisher=. 2018 , url=

    work page 2018

  53. [53]

    Trigonometric Series , subtitle=

    Antoni Zygmund , edition=. Trigonometric Series , subtitle=. 2003 , collection=

    work page 2003

  54. [54]

    Hardy and Werner W

    Godfrey H. Hardy and Werner W. Rogosinski , year=. Fourier Series , publisher=

    work page

  55. [55]

    Ul'yanov , title =

    Pyotr L. Ul'yanov , title =. Russian Mathematical Surveys , volume =. 1983 , url =

    work page 1983

  56. [56]

    Kolmogorov , title =

    Andrey N. Kolmogorov , title =. Fundamenta Mathematicae , volume =. 1923 , url =

    work page 1923

  57. [57]

    Kolmogorov , title =

    Andrey N. Kolmogorov , title =. Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences , volume =. 1926 , url =

    work page 1926

  58. [58]

    Studia Mathematica , volume =

    Yitzhak Katznelson , title =. Studia Mathematica , volume =. 1966 , url =

    work page 1966

  59. [59]

    Acta Mathematica , volume =

    Lennart Carleson , title =. Acta Mathematica , volume =. 1966 , url =

    work page 1966

  60. [60]

    Hunt , title =

    Richard A. Hunt , title =. Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) , publisher =

    work page 1967

  61. [61]

    Taylor , edition=

    Samuel Karlin and Howard M. Taylor , edition=. A First Course in Stochastic Processes , publisher=. 1975 , url=

    work page 1975

  62. [62]

    Silva , year=

    César E. Silva , year=. Invitation to Ergodic Theory , publisher=

    work page

  63. [63]

    I. P. Cornfeld and S. V. Fomin and Ya. G. Sinai , year=. Ergodic Theory , publisher=

    work page

  64. [64]

    Bradley , title =

    Richard C. Bradley , title =. Probability Surveys , volume =. 2005 , url =

    work page 2005

  65. [65]

    Mathematica Scandinavica , volume =

    Henry Helson and Donald Sarason , title =. Mathematica Scandinavica , volume =. 1967 , url =

    work page 1967

  66. [66]

    Wood , title =

    David C. Wood , title =. 1992 , url =

    work page 1992

  67. [67]

    Srivastava and Junesang Choi , year=

    Hari M. Srivastava and Junesang Choi , year=. Zeta and q -Zeta Functions and Associated Series and Integrals , publisher=

    work page

  68. [68]

    Physical Review E , volume =

    Zoltán Eisler and János Kertész , title =. Physical Review E , volume =. 2006 , url =

    work page 2006

  69. [69]

    Jiang and L

    Z.-Q. Jiang and L. Guo and W.-X. Zhou , title =. The European Physical Journal B , volume =. 2007 , url =

    work page 2007

  70. [70]

    Bickel and Peter Bühlmann , title =

    Peter J. Bickel and Peter Bühlmann , title =. Proceedings of the National Academy of Sciences of the United States of America , volume =. 1996 , url =

    work page 1996

  71. [71]

    Bickel and Peter Bühlmann , title =

    Peter J. Bickel and Peter Bühlmann , title =. Journal of Theoretical Probability , volume =. 1997 , url =

    work page 1997

  72. [72]

    Cox , title =

    David R. Cox , title =. Statistics: An Appraisal , editor =. 1984 , pages =

    work page 1984

  73. [73]

    Lithuanian Mathematical Journal , volume =

    Donata Puplinskaitė and Donatas Surgailis , title =. Lithuanian Mathematical Journal , volume =. 2009 , url =

    work page 2009

  74. [74]

    Advances in Applied Probability , volume =

    Donata Puplinskaitė and Donatas Surgailis , title =. Advances in Applied Probability , volume =. 2010 , url =

    work page 2010

  75. [75]

    Journal of Time Series Analysis , volume =

    Anne Philippe and Donata Puplinskaitė and Donatas Surgailis , title =. Journal of Time Series Analysis , volume =. 2013 , url =

    work page 2013

  76. [76]

    Bernoulli , publisher =

    Donata Puplinskaitė and Donatas Surgailis , title =. Bernoulli , publisher =. 2016 , url =

    work page 2016

  77. [77]

    Kendal and Bent J

    Wayne S. Kendal and Bent J. Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality , journal =. 2011 , url =

    work page 2011

  78. [78]

    Georganas , title =

    Boris Tsybakov and Nicolas D. Georganas , title =. IEEE/ACM Transactions on Networking , volume =. 1997 , doi=

    work page 1997

  79. [79]

    Self-Similar Network Traffic: An Overview , booktitle=

    Kihong Park and Walter Willinger , publisher=. Self-Similar Network Traffic: An Overview , booktitle=. 2000 , url=

    work page 2000

  80. [80]

    Taqqu and Will E

    Walter Willinger and Murad S. Taqqu and Will E. Leland and Daniel V. Wilson , title =. Statistical Science , publisher =. 1995 , doi=

    work page 1995

Showing first 80 references.