Pith. sign in

REVIEW 3 major objections 6 minor 42 references

Discrete-space and -time analogue of a super-diffusive fractional Brownian motion

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Two simple rounding rules turn a super-diffusive fractional Brownian motion into an integer-time lattice walk, preserving its power-law memory.

desk verdict A useful practical recipe for lattice super-diffusive fBm, but the numerical evidence undercuts the paper's central claim of exact covariance preservation. read the letter →

arxiv 2506.09921 v1 pith:3NGLMKNN submitted 2025-06-11 cond-mat.stat-mech cond-mat.dis-nnphysics.bio-phq-bio.BM

Enzo Marinari , Gleb Oshanin This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nnphysics.bio-phq-bio.BM MSC 60G2260G1582C31
keywords fractionalBrownianmotionGaussiannoisesuper-diffusionlatticerandomwalkpower-lawcorrelationsHurstexponentbinarysequencediscretizationanomalousdiffusion
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

Fractional Brownian motion (fBm) is the canonical Gaussian process with power-law memory, but its trajectories live in continuous space and time, which is awkward for lattice simulations and for any setting where steps must be integers. This paper proposes two elementary discretizations — taking only the sign of each increment (TLD), or rounding its magnitude to a positive integer while keeping the sign (FOD) — and argues numerically that both produce an integer-time random walk on the integers with the same power-law increment covariance as the parent fBm for Hurst exponents $H>1/2$. The claim matters because a faithful lattice analogue makes the super-diffusive regime accessible to exact numerics and to models built from correlated binary or integer sequences, while also exposing a structural difference from sub-diffusive fBm, for which no such analogue is achieved.

What carries the argument

The load-bearing identity is the increment covariance of fBm, $C(|t-t'|)=\langle\delta_t\delta_{t'}\rangle=H(2H-1)/|t-t'|^{2-2H}$, whose power-law exponent $\varepsilon=2-2H$ is the quantity that must survive discretization. The two algorithms are the machinery: TLD maps each increment $\delta_t$ to $\operatorname{sign}(\delta_t)$, while FOD maps it to $\pm\lfloor|\delta_t|+1/2\rfloor$ and rescales the effective time step by the realized walk length so that comparison with the continuous path is meaningful. Because the paper works with centered Gaussian processes, matching this second-order statistic fixes the process; the numerics check the reduced covariance at lags spanning three decades.

What would settle it

Simulate the TLD and FOD walks for $H=0.55$ and $H=0.95$ with $T=2^{22}$ and roughly 2000 realizations, fit the reduced increment covariance $\widetilde{C}(d\tau)$ over lags from 1 to 1000, and check whether the fitted exponents agree with $2-2H$ within error; a mismatch, or a strong dependence on the per-trajectory rescaling rule in FOD, would falsify the claimed universality.

Watch

Extended reading notes

Core claim

The central discovery is that a super-diffusive fBm with Hurst index $H>1/2$ survives a drastic discretization of its increments. If $\delta_t$ is an increment of a continuous fBm trajectory, the TLD rule replaces it by $+1$ or $-1$ according to its sign, and the FOD rule replaces it by $\pm\lfloor|\delta_t|+1/2\rfloor$ with a per-trajectory rescaling of the time step. Numerical fits for $H=0.6$, $0.7$, $0.8$, and $0.9$ show that the reduced increment covariance $\widetilde{C}(d\tau)=\langle\delta_t\delta_{t+d\tau}\rangle/\langle\delta_t\delta_{t+1}\rangle$ decays as $(d\tau)^{-\varepsilon}$ with $\varepsilon$ matching $2-2H$ for both schemes, and individual discretized trajectories follow the parent trajectory with Pearson coefficients above $0.999$. The FOD scheme is slightly closer to the parent process. The same construction fails for sub-diffusion, where correlations are integrable and short-time sign structure matters.

Load-bearing premise

The load-bearing premise is that replacing an fBm increment by its sign, or by its rounded magnitude, preserves the power-law decay exponent $\varepsilon=2-2H$ of the increment covariance for every $H>1/2$; the paper checks this only by finite-time fits at $H=0.6$, $0.7$, $0.8$, and $0.9$, and the FOD rescaling is applied separately to each trajectory, which could bias ensemble averages.

Editorial extensions

If this is right

  • For any $H>1/2$, one can generate an integer-time walk on $\mathbb{Z}$ whose increment correlations decay as $|t-t'|^{-(2-2H)}$, giving a ready-made lattice model for super-diffusive transport.
  • The TLD scheme produces a binary $\pm1$ sequence with power-law correlations, useful wherever correlated binary inputs are needed.
  • The FOD scheme is the more faithful of the two, nearly indistinguishable from the parent fBm covariance over three decades of lag.
  • The construction extends to $D$ dimensions by running independent copies per axis.
  • A lattice analogue of sub-diffusive fBm remains open, and the authors argue sub-diffusion is governed by non-universal short-time features rather than by long-range integrated correlations.

Reading between the lines

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

  • If the exponent preservation holds for all $H\in(1/2,1)$, one can view sign-thresholded fractional Gaussian noise as a constructive source of binary sequences with arbitrary tunable power-law correlations, potentially useful in coding and memory models; the paper only demonstrates this numerically for four values of $H$.
  • The per-realization time rescaling in FOD may bias ensemble covariance estimates; testing a global, ensemble-level rescaling would clarify whether the apparent advantage of FOD over TLD is intrinsic or an artifact of the normalization.
  • The near-perfect Pearson correlation between continuous and discretized individual trajectories suggests that, for $H>1/2$, the sign process carries almost all the macroscopic shape of the path; a formal bound on this linear dependence would turn the numerical observation into a theorem.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes two constructions of a discrete-space, integer-time random walk intended to serve as a lattice analogue of super-diffusive fractional Brownian motion (fBm) with Hurst exponent H>1/2. In the TLD scheme, increments are binarized by their sign; in the FOD scheme, increments are rounded in magnitude and the time axis is rescaled per trajectory via Eq. (6). The authors validate the constructions numerically using Davies-Harte generated fBm trajectories (2000 realizations, T=2^22, H=0.6-0.9): they report power-law fits for the increment covariance in Table I, show good collapse of the averaged absolute displacement in Fig. 1 and of the reduced covariance in Fig. 2, and report very high Pearson correlations between individual original and discretized trajectories. They conclude that TLD and FOD are discrete counterparts of super-diffusive fBm and argue that super-diffusion is universal while sub-diffusion is not.

Significance. If the central claim were established, the paper would provide simple, practical algorithms (O(T log T) using circulant embedding) for generating lattice walks with fBm-like increment correlations, and it would sharpen the distinction between super-diffusive persistence and sub-diffusive anti-persistence. The numerical effort is substantial and the algorithms are transparent, with TLD being essentially parameter-free. However, the central claim is currently overstated: the exact covariance equality asserted in the text is false for TLD, the numerical fits in Table I deviate from the theoretical exponent by many quoted error bars, and the FOD per-trajectory time rescaling undermines the integer-time interpretation. These issues are load-bearing because the paper's conclusion rests on the claim that the increment covariance 'obeys Eq. (4)' and that the fits show 'perfect agreement'. With a reformulation in terms of asymptotic tail equivalence and a more careful statistical analysis, the contribution could still be useful.

major comments (3)
  1. [Sec. IV, Table I] The fitted exponents do not agree with the theoretical exponent ε_th=2−2H within the quoted errors. For H=0.7, ε_TLD=0.628±0.004 versus ε_th=0.6, a 7σ deviation; for H=0.9, ε_TLD=0.225±0.002 versus 0.2 (12.5σ) and ε_FOD=0.211±0.001 (11σ). Even the parent fBm fit shows deviations (ε_X=0.838±0.008 versus 0.8 for H=0.6, and 0.206±0.001 versus 0.2 for H=0.9). Thus the statement that results agree 'in the limits of the statistical error' is not supported by the reported numbers. The authors should report the fitting range, the method used to estimate errors (e.g., bootstrap over realizations), and should either correct for the apparent finite-window bias or explicitly weaken the claim to asymptotic agreement.
  2. [Sec. III, TLD rule and Eq. (4)] For the TLD scheme, the increment covariance is not the same power law as Eq. (4). Since the continuous-time increments (δ_t, δ_{t+dτ}) are jointly Gaussian with correlation ρ(dτ)=|dτ|^{-(2-2H)} (up to normalization), the sign covariance is exactly E[sign(δ_t)sign(δ_{t+dτ})]=(2/π)arcsin(ρ(dτ)). This equals a constant times (dτ)^{-(2-2H)} only in the asymptotic limit ρ→0; over any finite lag window a power-law fit will produce an effective exponent larger than 2−2H, which is consistent with the positive deviations seen in Table I. Therefore the claim that TLD increments have 'exactly the same power-law covariance function' as fBm is incorrect. The authors should either replace this claim with the explicit arcsin formula and an asymptotic statement, or restrict the claim to asymptotic tail equivalence.
  3. [Sec. III, Eq. (6)] The FOD time rescaling is applied per realization: the factor T/T_FOD depends on the random variable T_FOD, the number of nonzero steps in that particular trajectory. This means each trajectory is reparametrized differently, so the ensemble-averaged covariance C̃(dτ) computed in Eq. (7) mixes the increment correlations with the distribution of T_FOD. Moreover, the rescaled time step τ'_FOD = τ × T/T_FOD is generally not an integer, so the FOD process is not an integer-time process in the sense claimed in the abstract and introduction. The authors should specify the ensemble procedure unambiguously (e.g., a fixed global rescaling or a conditional analysis) and clarify in what precise sense FOD is a discrete-time analogue.
minor comments (6)
  1. [Sec. IV] The text refers to 'FOB schemes' in the paragraph after Fig. 2; this should read 'FOD schemes'.
  2. [Table I] Table I lists H=0.6, 0.7, and 0.9 but omits H=0.8, which is the representative case used in Figs. 1-3; either include H=0.8 in the table or explain the omission.
  3. [Fig. 1] The caption says 'ensemble averaged absolute value of X(t)', but the text does not distinguish between ⟨|X(t)|⟩ and √⟨X^2(t)⟩; please state explicitly which quantity is plotted and why the mean squared displacement itself is not shown.
  4. [Sec. IV, Pearson coefficient] The Pearson values, e.g., '.999937', should be written as '0.999937'; also report the number of trajectories used to compute the coefficient and the statistical uncertainty, since the reported values differ only in the fourth decimal.
  5. [Eq. (7)] The reduced covariance is normalized by ⟨δ_t δ_{t+1}⟩; for the TLD and FOD processes this denominator is not equal to H(2H−1), so the comparison in Fig. 2 should state how each curve is normalized before dividing by the common factor.
  6. [Sec. IV] The sentence 'The method can be easily applied to D dimensional systems with D>1' is not accompanied by any test; either provide a D>1 demonstration or label this as a conjecture.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: TLD/FOD are deterministic functions of fBm increments and are validated against the parent covariance, but the match is not enforced by construction and no load-bearing self-citation is used.

full rationale

The paper's derivation chain is self-contained. It starts from the standard Mandelbrot–van Ness fBm definition (Eq. 1) and its known increment covariance (Eq. 4). The proposed TLD and FOD schemes (Sec. III, Eqs. 5–6) are deterministic nonlinear functions of the parent fBm increments: sign thresholding and magnitude rounding. The central claim in Sec. IV is that the covariance functions of the resulting lattice processes decay with the same exponent ε_th = 2−2H. This is verified numerically by fitting Eq. (9) to simulated data (Table I), not derived from the input by construction. The sign and rounding transformations do not preserve covariance exactly in general; indeed, for TLD the exact sign covariance would be (2/π)arcsin(ρ), which is not a pure power law. Thus the numerical agreement is a nontrivial empirical claim, not an identity. The normalization in Eq. (7) sets C̃(1)=1 for all processes, but that is an acknowledged by-construction normalization and does not determine the fitted exponents at dτ>1. The high Pearson coefficients in Sec. IV are expected consequences of applying monotone (sign-preserving) transformations to a strongly persistent process; the paper presents them as such, not as independent predictions. No load-bearing self-citation occurs: Refs. 26 and 27 are background citations for path-integral representations of fBm and are not used to justify the algorithms or the covariance claim. The reported fit discrepancies (e.g., ε_TLD=0.628±0.004 vs ε_th=0.6 for H=0.7) are a correctness/support concern, not a circularity, and are outside the scope of this pass. Therefore the paper does not reduce to its inputs by definition.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The discretization schemes introduce two hand-chosen elements: the FOD rounding offset and the per-trajectory time rescaling. The validation relies on standard fBm covariance theory and the Davies-Harte algorithm, plus an implicit assumption that covariance matching suffices for a non-Gaussian lattice analogue. No new entities are postulated.

free parameters (2)
  • FOD rounding offset = 1/2
    In Eq. (5), phi_t = floor(|delta_t| + 1/2). The 1/2 offset is chosen by hand; it sets the boundaries between step sizes and the condition for pauses (|delta_t| < 1/2). Different offsets yield different step-size distributions and could alter short-time correlations.
  • FOD per-trajectory time rescaling factor T/T_FOD = depends on each realization
    Eq. (6) rescales the time axis of each FOD walk individually to match the continuous trajectory's total time. This normalization is defined per trajectory and has no ensemble-level theoretical justification.
assumptions (4)
  • standard math fBm with Hurst exponent H is a centered Gaussian process with covariance (1/2)(t^{2H} + t'^{2H} - |t-t'|^{2H}) and increment covariance H(2H-1)|t-t'|^{-(2-2H)}.
    Used as the target definitions in Sec. II, Eqs. (1)-(4).
  • standard math The Davies-Harte/circulant algorithm generates trajectories with the exact finite-dimensional covariance of fBm.
    Invoked in Sec. IV to produce X(t) for discretization; correctness of the algorithm is taken from Refs. 36-38.
  • domain assumption Matching the increment covariance is a sufficient criterion for a lattice process to be an analogue of fBm.
    The paper states that Gaussian processes are defined by second-order statistics (Sec. IV), but the lattice processes are not Gaussian; the assumption that covariance equivalence suffices is implicit and not justified.
  • ad hoc to paper Super-diffusive fBm is universal while sub-diffusive fBm is governed by non-universal, local features.
    Inferred in Sec. V from the failure of the proposed discretizations for H<1/2. This is a speculative interpretation, not demonstrated by the data.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Discrete-space and -time analogue of a super-diffusive fractional Brownian motion." pith.science (2026). https://pith.science/paper/3NGLMKNN

@misc{pith2026250609921,
  author       = {Pith},
  title        = {Pith review of: Discrete-space and -time analogue of a super-diffusive fractional Brownian motion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NGLMKNN}},
  note         = {Machine review of arXiv:2506.09921}
}
read the original abstract

We discuss how to construct reliably well "a lattice and an integer time" version of a super-diffusive continuous-space and -time fractional Brownian motion (fBm) -- an experimentally-relevant non-Markovian Gaussian stochastic process with an everlasting power-law memory on the time-evolution of thermal noises extending over the entire past. We propose two algorithms, which are both validated by extensive numerical simulations showing that the ensuing lattice random walks have not only the same power-law covariance function as the standard fBm, but also individual trajectories follow those of the super-diffusive fBm. Finding a lattice and an integer time analogue of a sub-diffusion fBm, which is an anti-persistent process, remains a challenging open problem. Our results also clarify the relevant difference between sub-diffusive and super-diffusive fBm, that are frequently seen as two very analogous realizations of processes with memory. They are indeed substantially different.

Figures

Figures reproduced from arXiv: 2506.09921 by the authors.

Figure 1
Figure 1. FIG. 1. Ensemble averaged absolute value of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Log-log plot of the time-evolution of absolute values of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Barkai, Y

    E. Barkai, Y. Garini, and R. Metzler, Strange kinetics of single molecules in living cells , Phys. Today 65 , 29 (2012)

    work page 2012

  2. [2]

    S. S. Mogre, A. I. Brown, and E. F. Koslover, Getting around the cell: physical transport in the intracellular world , Phys. Biol. 17 , 061003 (2020)

    work page 2020

  3. [3]

    E. F. Koslover, Searching Through Cellular Landscapes , in: Target Search Problems, D. Grebenkov, R. Metzler and G. Oshanin, Eds., (Springer, Cham, 2024), p. 541

    work page 2024

  4. [4]

    Vilk et al., Unravelling the origins of anomalous diffusion: From molecules to migrating storks , Phys

    O. Vilk et al., Unravelling the origins of anomalous diffusion: From molecules to migrating storks , Phys. Rev. Research 4 , 033055 (2022)

    work page 2022

  5. [5]

    Bouchaud, A

    J.-P. Bouchaud, A. Comtet, A. Georges, and P. le Doussal, Classical diffusion of a particle in a one-dimensional random force field , Ann. Phys. 201 , 285 (1990)

    work page 1990

  6. [6]

    Bouchaud and A

    J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , Phys. Repts. 195 , 127 (1990)

    work page 1990

  7. [7]

    Oshanin and A

    G. Oshanin and A. Blumen, Rouse chain dynamics in layered random flows , Phys. Rev. E 49 , 4185 (1994)

    work page 1994

  8. [8]

    Squarcini, E

    A. Squarcini, E. Marinari, and G. Oshanin, Passive advection of fractional Brownian motion by random layered flows , New J. Phys. 22 , 053052 (2020)

    work page 2020

Show all 42 references
  1. [9]

    Mejia-Monasterio, S

    C. Mejia-Monasterio, S. Nechaev, G. Oshanin, and O. Vasilyev, Tracer diffusion on a crowded random Manhattan lattice , New J. Phys. 22 , 033024 (2020)

    2020

  2. [10]

    Scher and E

    H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids , Phys. Rev. B 12 , 2455 (1975)

    1975

  3. [11]

    Barkai, R

    E. Barkai, R. Metzler, and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation , Phys. Rev. E 61 , 132 (2000)

    2000

  4. [12]

    Gurtovenko and A

    A. Gurtovenko and A. Blumen, Generalized Gaussian Structures: Models for Polymer Systems with Complex Topologies , In: A. Abe, K. Dusek, and S. Kobayashi, Eds., Polymer Analysis Polymer Theory. Advances in Polymer Science, Vol. 182 , (Springer, Berlin, 2005)

    2005

  5. [13]

    Dolgushev, G

    M. Dolgushev, G. Berezovska, and A. Blumen, Branched Semiflexible Polymers: Theoretical and Simulation Aspects , Macromol. Theor. Simulations 20 , 621 (2011)

    2011

  6. [14]

    S. K. Ghosh, A. G. Cherstvy, D. S. Grebenkov, and R. Metzler, Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments , New J. Phys. 18 , 013027 (2016)

    2016

  7. [15]

    B\'enichou, P

    O. B\'enichou, P. Illien, G. Oshanin, A. Sarracino, and R. Voituriez, Tracer diffusion in crowded narrow channels , J. Phys.: Condens. Matter 30 , 443001 (2018)

    2018

  8. [16]

    Bender, Y

    C. Bender, Y. A. Butko, M. D'Ovidio and G. Pagnini, A Limit Theorem Clarifying the Physical Origin of Fractional Brownian Motion and Related Gaussian Models of Anomalous Diffusion, https://arxiv.org/abs/2411.18775

    arXiv

  9. [17]

    Metzler, J.-H

    R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking , Phys. Chem. Chem. Phys. 16 , 24128 (2014)

    2014

  10. [18]

    B. D. Hughes, Random Walks and Random Environments , Vol. 1, (Clarendon Press, Oxford, 1995)

    1995

  11. [19]

    Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium , Theor. Probab. Appl. 27 , 256 (1982)

    1982

  12. [20]

    Marinari, G

    E. Marinari, G. Parisi, D. Ruelle, and P. Windey, Random Walk in a Random Environment and 1/f Noise , Phys. Rev. Lett. 50 , 1223 (1983)

    1983

  13. [21]

    Metzler and J

    R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics , J. Phys. A: Math. Gen. 37 , R161 (2004)

    2004

  14. [22]

    Metzler and J

    R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach , Phys. Rep. 339 , 1 (2000)

    2000

  15. [23]

    Krapf et al., Spectral content of a single non-Brownian trajectory , Phys

    D. Krapf et al., Spectral content of a single non-Brownian trajectory , Phys. Rev. X 9 , 011019 (2019)

    2019

  16. [24]

    Sposini et al., Towards a robust criterion of anomalous diffusion , Commun Phys 5 , 305 (2022)

    V. Sposini et al., Towards a robust criterion of anomalous diffusion , Commun Phys 5 , 305 (2022)

    2022

  17. [25]

    Mandelbrot and J

    B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noises and applications , SIAM Review 10 , 422 (1968)

    1968

  18. [26]

    Meerson, O

    B. Meerson, O. B\'enichou, and G. Oshanin, Path integrals for fractional Brownian motion and fractional Gaussian noise , Phys. Rev. E 106 , L062102 (2022)

    2022

  19. [27]

    B\'enichou and G

    O. B\'enichou and G. Oshanin, A unifying representation of path integrals for fractional Brownian motions , J. Phys. A: Math. and Theor. 57 , 225001 (2024)

    2024

  20. [28]

    Jeon and R

    J.-H. Jeon and R. Metzler, Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries , Phys. Rev. E 81 , 021103 (2010)

    2010

  21. [29]

    D. J. Amit, H. Gutfreund and H. Sompolinsky, Statistical Mechanics of Neural Networks near Saturation , Ann. Phys. 173 , 30 (1987)

    1987

  22. [30]

    Czirok, R

    A. Czirok, R. N. Mantegna, S. Havlin and H. E. Stanley, Correlations in binary sequences and generalized Zipf analysis , Phys. Rev. E 52 , 446 (1995)

    1995

  23. [31]

    Levy, Random functions: General theory with special references to Laplacian random functions , University of California Publications in Statistics 1 , 331 (1953)

    P. Levy, Random functions: General theory with special references to Laplacian random functions , University of California Publications in Statistics 1 , 331 (1953)

    1953

  24. [32]

    A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum , C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 , 115 (1940)

    1940

  25. [33]

    Nourdin, Selected Aspects of Fractional Brownian Motion , (Springer & Bocconi University Press, Milano, 2012)

    I. Nourdin, Selected Aspects of Fractional Brownian Motion , (Springer & Bocconi University Press, Milano, 2012)

    2012

  26. [34]

    L. R. Evangelista and E. Kaminski Lenzi, Fractional Diffusion Equation and Anomalous Diffusion , (Cambridge University Press, Cambridge, 2018)

    2018

  27. [35]

    L. R. Evangelista and E. Kaminski Lenzi, An Introduction to Anomalous Diffusion and Relaxation , (Springer & Politecnico di Torino, Torino, 2023)

    2023

  28. [36]

    Dieker, Simulation of fractional Brownian motion , PhD Thesis, (University of Twente, Enschede, 2004)

    T. Dieker, Simulation of fractional Brownian motion , PhD Thesis, (University of Twente, Enschede, 2004)

    2004

  29. [37]

    A. B. Dieker and M. Mandjes, On spectral simulation of fractional Brownian motion , Probab. Eng. Inf. Sci. 17 , 417 (2003)

    2003

  30. [38]

    Davies and D.S

    R.B. Davies and D.S. Harte, Tests for Hurst effect , Biometrika 74 , 95 (1987)

    1987

  31. [39]

    A. T. A. Wood and G. Chan, Simulation of stationary Gaussian processes in [0, 1]^d , J. Comput. Graph. Stat. 3 , 409 (1994)

    1994

  32. [40]

    Dietrich and G.N

    C.R. Dietrich and G.N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix , SIAM Journal Sci. Comput. 18 , 1088 (1997)

    1997

  33. [41]

    Chan and A

    G. Chan and A. T. A. Wood, Simulation of stationary Gaussian vector fields , Stat. Comput. 9 , 265 (1999)

    1999

  34. [42]

    Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resour

    J.R.M. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resour. Res. 20 , 1898 (1984)

    1984

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.