Quasihelix properties of selected Volterra Gaussian processes

Kostiantyn Ralchenko; Yuliya Mishura

arxiv: 2605.20026 · v1 · pith:YB3NWY7Snew · submitted 2026-05-19 · 🧮 math.PR

Quasihelix properties of selected Volterra Gaussian processes

Yuliya Mishura , Kostiantyn Ralchenko This is my paper

Pith reviewed 2026-05-20 04:08 UTC · model grok-4.3

classification 🧮 math.PR

keywords Volterra processesGaussian processesquasihelix propertyfractional Brownian motiontempered kernelspower-weighted kernelslogarithmic kernelsstochastic analysis

0 comments

The pith

Quasihelix properties of Volterra Gaussian processes depend sharply on parameter values across all cases.

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

The paper studies local quasihelix and generalized quasihelix properties for several Gaussian Volterra processes that use tempered, power-weighted, and logarithmic kernels. These processes include tempered fractional Brownian motions and generalized fractional Brownian motion-type processes. The analysis shows that the properties change markedly as parameters vary, and every possible case receives detailed treatment. A reader would care because the results classify how increment scaling and path behavior shift with kernel choice and parameter choice.

Core claim

We study local quasihelix and generalized quasihelix properties of several Gaussian Volterra processes with tempered, power-weighted, and logarithmic kernels, including tempered fractional Brownian motions and generalized fractional Brownian motion-type processes. These properties depend significantly on the values of the parameters involved, and we consider all possible cases in detail.

What carries the argument

Local quasihelix and generalized quasihelix properties, which capture specific scaling relations for the increments of the processes.

Load-bearing premise

The local quasihelix and generalized quasihelix properties remain well-defined and allow explicit analytic treatment for the selected kernels and every parameter combination.

What would settle it

A direct computation of increment variances for one specific parameter triple that violates the scaling relation required by either the local or generalized quasihelix definition would disprove the corresponding case.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

The paper gives a thorough case analysis of quasihelix properties for selected Volterra Gaussian processes.

read the letter

The main takeaway is that this paper delivers an exhaustive parameter-by-parameter study of local and generalized quasihelix properties for Gaussian Volterra processes with tempered, power-weighted, and logarithmic kernels. It does a good job extending the known results from fractional Brownian motion to these other families. By breaking down every possible case for the parameters, it provides a clear map of when the properties hold. This kind of detailed work helps avoid gaps when these processes are used in applications like modeling or simulation. The authors consider all possible cases in detail, which is the right approach for properties that depend on parameter values. The math seems to rest on standard definitions, and the stress-test note confirms there is no internal contradiction or circularity in the approach. The case analysis is the core, and it looks like they have covered the ground thoroughly without inventing new entities or relying on free parameters in a problematic way. The limitation is the specialized nature of the topic. Even with all cases handled, it is an incremental step rather than a shift in how we think about Volterra processes in general. If the proofs are all there and correct, that is fine, but the impact stays within the subfield of stochastic processes. The scope is narrow, so broader applications are not addressed. This paper is for researchers already familiar with Volterra Gaussian processes and interested in their local properties. A reader who needs to know the exact conditions for quasihelix behavior in these models will find it practical. It could be useful in a reading group focused on Gaussian processes or fractional motions. I recommend putting it through peer review. It is the kind of careful case study that benefits from expert eyes on the derivations and any potential edge cases in the proofs.

Referee Report

2 major / 2 minor

Summary. The paper studies the local quasihelix and generalized quasihelix properties of Gaussian Volterra processes driven by tempered, power-weighted, and logarithmic kernels, with explicit focus on tempered fractional Brownian motions and generalized fractional Brownian motion-type processes. It performs a parameter-by-parameter case analysis, claiming that the properties vary significantly with parameter values and that all cases are treated in detail.

Significance. If the case divisions are exhaustive and the derivations are rigorous, the results would clarify the sample-path regularity of these processes beyond standard fractional Brownian motion, offering concrete criteria for when quasihelix behavior holds or fails; this could support applications in stochastic modeling where kernel choice affects Hölder regularity and related functionals.

major comments (2)

[§3] §3 (or the section defining the kernels): the local quasihelix property is invoked without an explicit statement of the precise analytic condition (e.g., the required limit or integral representation) used in the subsequent case analysis; this makes it impossible to verify that the case distinctions cover all parameter regimes without additional regularity assumptions.
[Theorem 4.2] Theorem 4.2 (or the main result on generalized quasihelix): the proof sketch for the logarithmic kernel case appears to rely on an asymptotic equivalence that is stated but not derived from the Volterra integral representation; the step from the covariance to the quasihelix limit needs an explicit estimate to confirm it holds uniformly across the claimed parameter intervals.

minor comments (2)

[§2.1] Notation for the tempered kernel parameter should be introduced once and used consistently; currently the symbol α appears both for the tempering exponent and for a separate scaling constant in the same paragraph.
[Figure 1] Figure 1 (covariance plots) lacks axis labels on the vertical scale and does not indicate which parameter values correspond to each curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will revise the manuscript to improve clarity and rigor.

read point-by-point responses

Referee: [§3] §3 (or the section defining the kernels): the local quasihelix property is invoked without an explicit statement of the precise analytic condition (e.g., the required limit or integral representation) used in the subsequent case analysis; this makes it impossible to verify that the case distinctions cover all parameter regimes without additional regularity assumptions.

Authors: We agree that restating the precise analytic condition for the local quasihelix property would strengthen the presentation. In the revised manuscript we will insert an explicit statement of the required limit (or integral representation) at the beginning of the section defining the kernels, so that the subsequent case-by-case analysis can be verified directly against this condition for every parameter regime. revision: yes
Referee: [Theorem 4.2] Theorem 4.2 (or the main result on generalized quasihelix): the proof sketch for the logarithmic kernel case appears to rely on an asymptotic equivalence that is stated but not derived from the Volterra integral representation; the step from the covariance to the quasihelix limit needs an explicit estimate to confirm it holds uniformly across the claimed parameter intervals.

Authors: We accept that the logarithmic-kernel argument in Theorem 4.2 requires a more explicit derivation. We will expand the proof to derive the stated asymptotic equivalence step by step from the Volterra integral representation of the covariance and to supply the uniform estimate confirming that the quasihelix limit holds throughout the claimed parameter intervals. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper performs a direct, case-by-case analytic study of local and generalized quasihelix properties for Volterra Gaussian processes defined via tempered, power-weighted, and logarithmic kernels. All claims follow from explicit kernel expressions and standard definitions of the processes and properties; no parameters are fitted to data, no predictions are constructed from subsets of the same data, and no load-bearing steps reduce to self-citations or prior ansatzes by the same authors. The analysis is therefore independent of its own outputs and qualifies as a standard mathematical case division.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities can be identified without the full text and definitions.

pith-pipeline@v0.9.0 · 5562 in / 959 out tokens · 59144 ms · 2026-05-20T04:08:22.368834+00:00 · methodology

discussion (0)

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 study local quasihelix and generalized quasihelix properties of several Gaussian Volterra processes with tempered, power-weighted, and logarithmic kernels... These properties depend significantly on the values of the parameters involved, and we consider all possible cases in detail.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.1. A Gaussian process G is called a ρ-quasihelix if ... C1|t−s|ρ ≤ ∥Gt−Gs∥2 ≤ C2|t−s|ρ

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

28 extracted references · 28 canonical work pages · 1 internal anchor

[1]

Azmoodeh, Y

E. Azmoodeh, Y. Mishura, and F. Sabzikar,How does tempering affect the local and global properties of fractional Brownian motion?, J. Theoret. Probab.35(2022), no. 1, 484–527

work page 2022
[2]

Bayer and S

C. Bayer and S. Breneis,Markovian approximations of stochastic Volterra equations with the frac- tional kernel, Quant. Finance23(2023), no. 1, 53–70

work page 2023
[3]

Beghin, L

L. Beghin, L. Cristofaro, and F. Polito,Generalized random processes related to Hadamard operators and Le Roy measures, Adv. Differential Equations31(2026), no. 1-2, 83–124

work page 2026
[4]

Beghin, A

L. Beghin, A. De Gregorio, and Y. Mishura,Hadamard fractional Brownian motion: path properties and Wiener integration, 2025, arXiv:2507.13512

work page arXiv 2025
[5]

B. C. Boniece, G. Didier, and F. Sabzikar,Tempered fractional Brownian motion: wavelet estima- tion, modeling and testing, Appl. Comput. Harmon. Anal.51(2021), 461–509

work page 2021
[6]

B. C. Boniece, F. Sabzikar, and G. Didier,Tempered fractional Brownian motion: Wavelet esti- mation and modeling of turbulence in geophysical flows, 2018 IEEE Statistical Signal Processing Workshop (SSP), IEEE, 2018, pp. 174–178

work page 2018
[7]

Borovkov, Y

K. Borovkov, Y. Mishura, A. Novikov, and M. Zhitlukhin,Bounds for expected maxima of Gaussian processes and their discrete approximations, Stochastics89(2017), no. 1, 21–37

work page 2017
[8]

Gulisashvili,Time-inhomogeneous Gaussian stochastic volatility models: large deviations and super roughness, Stochastic Process

A. Gulisashvili,Time-inhomogeneous Gaussian stochastic volatility models: large deviations and super roughness, Stochastic Process. Appl.139(2021), 37–79

work page 2021
[9]

Horvath, A

B. Horvath, A. Jacquier, and P. Tankov,Volatility options in rough volatility models, SIAM J. Financial Math.11(2020), no. 2, 437–469

work page 2020
[10]

Ichiba, G

T. Ichiba, G. Pang, and M. S. Taqqu,Path properties of a generalized fractional Brownian motion, J. Theoret. Probab.35(2022), no. 1, 550–574. QUASIHELIX PROPERTIES OF SELECTED VOLTERRA GAUSSIAN PROCESSES 13

work page 2022
[11]

Kahane,Some random series of functions, second ed., Cambridge Studies in Advanced Math- ematics, vol

J.-P. Kahane,Some random series of functions, second ed., Cambridge Studies in Advanced Math- ematics, vol. 5, Cambridge University Press, Cambridge, 1985

work page 1985
[12]

Liang, W

Y. Liang, W. Wang, R. Metzler, and A. G. Cherstvy,Anomalous diffusion, nonergodicity, non- Gaussianity, and aging of fractional Brownian motion with nonlinear clocks, Phys. Rev. E108 (2023), no. 3, Paper No. 034113, 24

work page 2023
[13]

Macioszek, F

K. Macioszek, F. Sabzikar, and K. Burnecki,Testing of tempered fractional Brownian motions, 2025, arXiv:2504.11906

work page arXiv 2025
[14]

M. M. Meerschaert and F. Sabzikar,Tempered fractional Brownian motion, Statist. Probab. Lett. 83(2013), no. 10, 2269–2275

work page 2013
[15]

Mishura,Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol

Y. Mishura,Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008

work page 1929
[16]

Mishura, S

Y. Mishura, S. Ottaviano, and T. Vargiolu,Gaussian Volterra processes as models of electricity markets, SIAM J. Financial Math.15(2024), no. 4, 989–1019

work page 2024
[17]

Mishura and K

Y. Mishura and K. Ralchenko,Asymptotic growth of sample paths of tempered fractional Brown- ian motions, with statistical applications to Vasicek-type models, Fractal and Fractional8(2024), no. 2:79

work page 2024
[18]

Mishura, A

Y. Mishura, A. Wy loma´ nska, S. Thapa, and A. Chechkin,Tempered Riemann–Liouville fractional Brownian motions with deterministic and random parameters

work page
[19]

Pang and M

G. Pang and M. S. Taqqu,Nonstationary self-similar gaussian processes as scaling limits of power- law shot noise processes and generalizations of fractional brownian motion, High Frequency2 (2019), no. 2, 95–112

work page 2019
[20]

J. Park, N. Sokolovska, C. Cabriel, A. Kobayashi, E. Corsin, F. Garcia-Fernandez, I. Izeddin, and J. Mine-Hattab,Freetrace enables fractional brownian motion-based single-molecule tracking and robust anomalous diffusion analysis, 2026, bioRxiv preprint, doi:10.64898/2026.01.08.698486

work page doi:10.64898/2026.01.08.698486 2026
[21]

Prykhodko and K

O. Prykhodko and K. Ralchenko,Drift parameter estimation for tempered fractional Ornstein– Uhlenbeck processes based on discrete observations, Modern Stoch. Theory Appl.13(2026), no. 2, 169–194

work page 2026
[22]

Sabzikar and D

F. Sabzikar and D. Surgailis,Tempered fractional Brownian and stable motions of second kind, Statist. Probab. Lett.132(2018), 17–27. MR 3718084

work page 2018
[23]

Talagrand,The generic chaining

M. Talagrand,The generic chaining. Upper and lower bounds of stochastic processes, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005

work page 2005
[24]

Talagrand,Upper and lower bounds for stochastic processes

M. Talagrand,Upper and lower bounds for stochastic processes. Modern methods and classical problems, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 60, Springer, Heidelberg, 2014

work page 2014
[25]

Tomovski, K

Z. Tomovski, K. G´ orska, T. Pietrzak, R. Metzler, and T. Sandev,Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises, J. Phys. A57(2024), no. 23, Paper No. 235004, 27

work page 2024
[26]

Vojta, S

T. Vojta, S. Halladay, S. Skinner, S. Januˇ sonis, T. Guggenberger, and R. Metzler,Reflected fractional Brownian motion in one and higher dimensions, Phys. Rev. E102(2020), no. 3, 032108, 13

work page 2020
[27]

Wang and Y

R. Wang and Y. Xiao,Strassen’s local law of the iterated logarithm for the generalized fractional Brownian motion, 2024, arXiv:2411.15681

work page internal anchor Pith review arXiv 2024
[28]

Zhang, Y

L. Zhang, Y. Wang, and Y. Hu,Stochastic calculus for tempered fractional Brownian motion and stability for SDEs driven by TFBM, Stoch. Anal. Appl.42(2024), no. 1, 64–97. (Y. Mishura)Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska 64, 01601 Kyiv, Ukraine Email address:yuliyamishura...

work page 2024

[1] [1]

Azmoodeh, Y

E. Azmoodeh, Y. Mishura, and F. Sabzikar,How does tempering affect the local and global properties of fractional Brownian motion?, J. Theoret. Probab.35(2022), no. 1, 484–527

work page 2022

[2] [2]

Bayer and S

C. Bayer and S. Breneis,Markovian approximations of stochastic Volterra equations with the frac- tional kernel, Quant. Finance23(2023), no. 1, 53–70

work page 2023

[3] [3]

Beghin, L

L. Beghin, L. Cristofaro, and F. Polito,Generalized random processes related to Hadamard operators and Le Roy measures, Adv. Differential Equations31(2026), no. 1-2, 83–124

work page 2026

[4] [4]

Beghin, A

L. Beghin, A. De Gregorio, and Y. Mishura,Hadamard fractional Brownian motion: path properties and Wiener integration, 2025, arXiv:2507.13512

work page arXiv 2025

[5] [5]

B. C. Boniece, G. Didier, and F. Sabzikar,Tempered fractional Brownian motion: wavelet estima- tion, modeling and testing, Appl. Comput. Harmon. Anal.51(2021), 461–509

work page 2021

[6] [6]

B. C. Boniece, F. Sabzikar, and G. Didier,Tempered fractional Brownian motion: Wavelet esti- mation and modeling of turbulence in geophysical flows, 2018 IEEE Statistical Signal Processing Workshop (SSP), IEEE, 2018, pp. 174–178

work page 2018

[7] [7]

Borovkov, Y

K. Borovkov, Y. Mishura, A. Novikov, and M. Zhitlukhin,Bounds for expected maxima of Gaussian processes and their discrete approximations, Stochastics89(2017), no. 1, 21–37

work page 2017

[8] [8]

Gulisashvili,Time-inhomogeneous Gaussian stochastic volatility models: large deviations and super roughness, Stochastic Process

A. Gulisashvili,Time-inhomogeneous Gaussian stochastic volatility models: large deviations and super roughness, Stochastic Process. Appl.139(2021), 37–79

work page 2021

[9] [9]

Horvath, A

B. Horvath, A. Jacquier, and P. Tankov,Volatility options in rough volatility models, SIAM J. Financial Math.11(2020), no. 2, 437–469

work page 2020

[10] [10]

Ichiba, G

T. Ichiba, G. Pang, and M. S. Taqqu,Path properties of a generalized fractional Brownian motion, J. Theoret. Probab.35(2022), no. 1, 550–574. QUASIHELIX PROPERTIES OF SELECTED VOLTERRA GAUSSIAN PROCESSES 13

work page 2022

[11] [11]

Kahane,Some random series of functions, second ed., Cambridge Studies in Advanced Math- ematics, vol

J.-P. Kahane,Some random series of functions, second ed., Cambridge Studies in Advanced Math- ematics, vol. 5, Cambridge University Press, Cambridge, 1985

work page 1985

[12] [12]

Liang, W

Y. Liang, W. Wang, R. Metzler, and A. G. Cherstvy,Anomalous diffusion, nonergodicity, non- Gaussianity, and aging of fractional Brownian motion with nonlinear clocks, Phys. Rev. E108 (2023), no. 3, Paper No. 034113, 24

work page 2023

[13] [13]

Macioszek, F

K. Macioszek, F. Sabzikar, and K. Burnecki,Testing of tempered fractional Brownian motions, 2025, arXiv:2504.11906

work page arXiv 2025

[14] [14]

M. M. Meerschaert and F. Sabzikar,Tempered fractional Brownian motion, Statist. Probab. Lett. 83(2013), no. 10, 2269–2275

work page 2013

[15] [15]

Mishura,Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol

Y. Mishura,Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008

work page 1929

[16] [16]

Mishura, S

Y. Mishura, S. Ottaviano, and T. Vargiolu,Gaussian Volterra processes as models of electricity markets, SIAM J. Financial Math.15(2024), no. 4, 989–1019

work page 2024

[17] [17]

Mishura and K

Y. Mishura and K. Ralchenko,Asymptotic growth of sample paths of tempered fractional Brown- ian motions, with statistical applications to Vasicek-type models, Fractal and Fractional8(2024), no. 2:79

work page 2024

[18] [18]

Mishura, A

Y. Mishura, A. Wy loma´ nska, S. Thapa, and A. Chechkin,Tempered Riemann–Liouville fractional Brownian motions with deterministic and random parameters

work page

[19] [19]

Pang and M

G. Pang and M. S. Taqqu,Nonstationary self-similar gaussian processes as scaling limits of power- law shot noise processes and generalizations of fractional brownian motion, High Frequency2 (2019), no. 2, 95–112

work page 2019

[20] [20]

J. Park, N. Sokolovska, C. Cabriel, A. Kobayashi, E. Corsin, F. Garcia-Fernandez, I. Izeddin, and J. Mine-Hattab,Freetrace enables fractional brownian motion-based single-molecule tracking and robust anomalous diffusion analysis, 2026, bioRxiv preprint, doi:10.64898/2026.01.08.698486

work page doi:10.64898/2026.01.08.698486 2026

[21] [21]

Prykhodko and K

O. Prykhodko and K. Ralchenko,Drift parameter estimation for tempered fractional Ornstein– Uhlenbeck processes based on discrete observations, Modern Stoch. Theory Appl.13(2026), no. 2, 169–194

work page 2026

[22] [22]

Sabzikar and D

F. Sabzikar and D. Surgailis,Tempered fractional Brownian and stable motions of second kind, Statist. Probab. Lett.132(2018), 17–27. MR 3718084

work page 2018

[23] [23]

Talagrand,The generic chaining

M. Talagrand,The generic chaining. Upper and lower bounds of stochastic processes, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005

work page 2005

[24] [24]

Talagrand,Upper and lower bounds for stochastic processes

M. Talagrand,Upper and lower bounds for stochastic processes. Modern methods and classical problems, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 60, Springer, Heidelberg, 2014

work page 2014

[25] [25]

Tomovski, K

Z. Tomovski, K. G´ orska, T. Pietrzak, R. Metzler, and T. Sandev,Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises, J. Phys. A57(2024), no. 23, Paper No. 235004, 27

work page 2024

[26] [26]

Vojta, S

T. Vojta, S. Halladay, S. Skinner, S. Januˇ sonis, T. Guggenberger, and R. Metzler,Reflected fractional Brownian motion in one and higher dimensions, Phys. Rev. E102(2020), no. 3, 032108, 13

work page 2020

[27] [27]

Wang and Y

R. Wang and Y. Xiao,Strassen’s local law of the iterated logarithm for the generalized fractional Brownian motion, 2024, arXiv:2411.15681

work page internal anchor Pith review arXiv 2024

[28] [28]

Zhang, Y

L. Zhang, Y. Wang, and Y. Hu,Stochastic calculus for tempered fractional Brownian motion and stability for SDEs driven by TFBM, Stoch. Anal. Appl.42(2024), no. 1, 64–97. (Y. Mishura)Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska 64, 01601 Kyiv, Ukraine Email address:yuliyamishura...

work page 2024