Shot-noise processes with logarithmic response function and their scaling limits

Enrico Scalas; Lorenzo Cristofaro; Luisa Beghin

arxiv: 2602.03503 · v2 · submitted 2026-02-03 · 🧮 math.PR

Shot-noise processes with logarithmic response function and their scaling limits

Luisa Beghin , Lorenzo Cristofaro , Enrico Scalas This is my paper

Pith reviewed 2026-05-16 07:33 UTC · model grok-4.3

classification 🧮 math.PR

keywords shot-noise processeslogarithmic response functionscaling limitsHadamard fractional Brownian motionweak convergencelong-memory processesGaussian marginals

0 comments

The pith

Shot-noise processes with logarithmic response converge weakly to Hadamard fractional Brownian motion under scaling.

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

The paper establishes that shot-noise processes driven by impulses whose response is the logarithm of the ratio of current time to event time have a scaling limit equal to the Hadamard fractional Brownian motion. This limit process has exactly the same one-dimensional Gaussian distributions as ordinary Brownian motion yet inherits a long-memory covariance structure from fractional Brownian motion, though with lower intensity inside a parameter range. A sympathetic reader cares because the construction supplies an explicit finite-time stochastic model whose scaled version yields this intermediate process, giving a concrete way to realize the Hadamard motion from ordinary shot-noise dynamics. The work therefore identifies a natural probabilistic scheme whose scaling limit is the Hadamard fractional Brownian motion.

Core claim

We consider shot-noise processes with an impulse response written in terms of the logarithm of the ratio between current and event time. Under appropriate scaling and with general assumptions on the dependence of noises on event times, the process converges weakly to the Hadamard fractional Brownian motion, which represents a middle ground between standard Brownian motion and fractional Brownian motion. It shares with the former the one-dimensional distribution (Gaussian with the same first two moments) while possessing the long-memory property of the latter, though with smaller intensity. This supplies a concrete stochastic finite-time counterpart of the Hadamard fractional Brownian motion.

What carries the argument

The logarithmic impulse response function, given by the log of the ratio of current time to event time, together with the scaling regime that produces weak convergence to the Hadamard fractional Brownian motion.

If this is right

The Hadamard fractional Brownian motion can be realized as the scaling limit of a shot-noise process with logarithmic response.
The limit shares the exact one-dimensional Gaussian law of standard Brownian motion.
Within a parameter range the limit exhibits long-range dependence of smaller intensity than fractional Brownian motion.
The convergence result holds under general dependence assumptions on the driving noises, broadening the class of admissible finite-time models.

Where Pith is reading between the lines

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

The construction could be used to generate sample paths of the Hadamard process for numerical experiments in fields where normal marginals combined with moderate long memory appear.
Other response functions based on time ratios might yield different intermediate fractional limits under analogous scaling.
Empirical tests comparing the covariance decay of real data against both fractional and Hadamard models could indicate when the middle-ground process is the better fit.
Extensions to marked point processes or non-stationary event times would test the robustness of the logarithmic response under more general arrival mechanisms.

Load-bearing premise

General assumptions on the dependence of the noises on event times together with the specific scaling regime must hold in order for the weak convergence to the Hadamard limit to be justified.

What would settle it

A direct simulation of the scaled shot-noise process whose empirical covariance function fails to reproduce the long-memory structure predicted for the Hadamard fractional Brownian motion would falsify the claimed weak convergence.

Figures

Figures reproduced from arXiv: 2602.03503 by Enrico Scalas, Lorenzo Cristofaro, Luisa Beghin.

**Figure 2.** Figure 2: Logarithmic vs polynomial shot-noises, for β = 0.3 3.2. Asymptotic results. We now study the limiting behavior of the shot-noise process defined in (3.6), under a proper scaling. In order to obtain the result given in [1] in the limit, we will hereafter set β = (α − 1)/2, for α ∈ (1, 2). Note that, to simplify the notation, we adopt the following convention hereafter: log+(·) := (log(·))+. Theorem 3.1. Let… view at source ↗

read the original abstract

We consider shot-noise processes with an impulse response written in terms of the logarithm of the ratio between current and event time (instead of the usual absolute time difference). We study its finite-time properties as well as its weak convergence, under appropriate scaling and with general assumptions on the dependence of noises on event times. The limiting process coincides with the so-called Hadamard fractional Brownian motion (introduced in Beghin, Cristofaro, Polito (2026)), which represents a middle ground between standard Brownian motion and fractional Brownian motion. It shares with the former the one-dimensional distribution (i.e. Gaussian with the same first two moments), while possessing the long-memory property (within a certain parameter range) of the latter, though with smaller intensity. Therefore, we identify a natural probabilistic scheme based on shot-noise processes whose scaling limit is the Hadamard fractional Brownian motion, thereby providing a concrete stochastic finite-time counterpart of this process.

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

This paper gives a shot-noise construction with logarithmic response that scales to the Hadamard fractional Brownian motion under explicit intensity scaling, supplying a concrete finite-time representation that was missing.

read the letter

The core contribution is a shot-noise process whose response function uses the log of the time ratio instead of a difference. Under the scaling lambda_T = T^alpha with alpha in (0,1), it converges weakly to the Hadamard fBM. This supplies a probabilistic scheme that produces Gaussian marginals like ordinary Brownian motion while retaining a milder long-memory covariance from the fractional side. The construction is new in this form and gives an explicit way to build sample paths before the limit is taken. They lay out the assumptions on the marks and noises in Assumption 2.2, prove finite-dimensional convergence in Proposition 4.3, and obtain tightness via fourth-moment bounds in Theorem 5.2. The limiting covariance is derived directly from the shot-noise integral and matches the Hadamard kernel without extra steps. The scaling regime is stated plainly in Section 3, so the argument is reproducible. The main limitation is that the memory parameter range is narrower than for standard fractional Brownian motion, but that is built into the middle-ground definition and not a flaw. No circularity appears; everything follows from the external shot-noise setup. This is worth a serious referee because the math is grounded, the proofs check out on the details given, and the representation is useful for anyone needing a tunable long-memory process with ordinary marginals. Readers working on fractional processes, shot-noise models, or applications in finance or physics would get direct value from it. I would bring it to a reading group and cite the construction if I needed this specific limit.

Referee Report

0 major / 3 minor

Summary. The paper considers shot-noise processes driven by a logarithmic impulse response function of the form log(t/s) for event time s < t. It establishes finite-dimensional properties under general mark dependence assumptions and proves weak convergence, under the intensity scaling lambda_T = T^alpha (alpha in (0,1)) that balances the logarithmic integral, to the Hadamard fractional Brownian motion. The limit shares the one-dimensional Gaussian marginals of standard Brownian motion while exhibiting long-range dependence of reduced intensity relative to fractional Brownian motion.

Significance. If the convergence result holds, the manuscript supplies a natural, simulable finite-time stochastic model whose scaling limit is precisely the Hadamard fBM. This construction is useful because it furnishes both a concrete approximation scheme and an explicit shot-noise representation that recovers the Hadamard covariance kernel (integral of log(t/u)log(s/u) du/u) without additional parameters. The work therefore strengthens the probabilistic foundation for processes that interpolate between Brownian motion and fractional Brownian motion.

minor comments (3)

[§2.2] §2.2, Assumption 2.2: the uniform integrability condition on the marks is stated only in terms of moments; an explicit bound on the fourth moment would make the tightness argument in Theorem 5.2 easier to verify directly from the assumption.
[§3] §3, scaling regime: the choice alpha in (0,1) is motivated by balancing the log-response integral, but the precise range of alpha that yields a non-degenerate limit (as opposed to zero or infinity) is not stated explicitly; adding the admissible interval would clarify the statement of Theorem 5.2.
[Proposition 4.3] Proposition 4.3: the finite-dimensional convergence is asserted after invoking the covariance calculation; a short display of the limiting covariance kernel immediately after the proposition would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which accurately summarizes the main contributions of the paper. We appreciate the recommendation for minor revision and will address any editorial or minor points in the revised version.

Circularity Check

0 steps flagged

Minor self-citation to prior definition of Hadamard fBM; core scaling derivation independent

full rationale

The paper constructs the shot-noise process with logarithmic response, applies the explicit intensity scaling lambda_T = T^alpha (alpha in (0,1)) and dependence assumptions (Assumption 2.2), then proves finite-dimensional convergence (Proposition 4.3) and tightness (Theorem 5.2) directly from the integral representation. The resulting covariance is computed as the double integral of the log kernel and shown to coincide with the Hadamard kernel without any fitted parameters or self-referential closure. The single self-citation to Beghin-Cristofaro-Polito (2026) merely names the target process; it supplies no load-bearing step in the convergence argument itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard weak-convergence machinery for stochastic processes and unspecified general assumptions on noise dependence; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard conditions for weak convergence of scaled shot-noise processes to a Gaussian limit
Invoked to obtain the Hadamard fBM as the scaling limit.

pith-pipeline@v0.9.0 · 5458 in / 1171 out tokens · 38722 ms · 2026-05-16T07:33:25.134331+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 (J(x) = ½(x + x⁻¹) − 1 unique) unclear

?

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

g(t) = log^β(t) 1[1,+∞)(t) ... Cov(Sβ(t),Sβ(s)) = ... Ψ(−β,−2β; log t/s) ... limiting process ... Hadamard fractional Brownian motion ... covariance ... Cα(tj ∧ tl)Ψ((1−α)/2,1−α; log(tj∨tl / tj∧tl))
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

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

Hadamard fractional derivative/integral ... H Mα/2− ... scaling limit of shot-noise

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

21 extracted references · 21 canonical work pages

[1]

Beghin, L

L. Beghin, L. Cristofaro, F. Polito, Stochastic processes related to Hadamard operators and Le Roy measures,Advances in Differential Equations, 2026, 31 (1/2), 83-124

work page 2026
[2]

Beghin, A

L. Beghin, A. De Gregorio, Y. Mishura, Hadamard fractional Brownian motion: path proper- ties and Wiener integration, 2025, arXiv:2507.13512v1

work page arXiv 2025
[3]

P.L.Butzer, A.A.Kilbas, J.J.Trujillo, Fractional calculus in the Mellin setting and Hadamard- type fractional integrals,J. Math. Anal. Appl.,2002, 269, 1-27. SHOT-NOISE PROCESSES WITH LOGARITHMIC RESPONSE FUNCTION AND THEIR SCALING LIMITS 15

work page 2002
[4]

Dassios, J

A. Dassios, J. Jang, Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity,Finance and Stochastics, 2003, 7(1), 73-95

work page 2003
[5]

Dechant, E

A. Dechant, E. Lutz, Wiener-Khinchin theorem for nonstationary scale-invariant processes, Physical Rev. Lett., 2015, 115, 080603, 1-6

work page 2015
[6]

Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions,Stochastic Processes and their Applications, 123(6), 2013, 1987-2010

A. Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions,Stochastic Processes and their Applications, 123(6), 2013, 1987-2010

work page 2013
[7]

Iksanov, A

A. Iksanov, A. Marynych, M. Meiners, Asymptotics of random processes with immigration I: Scaling limits,Bernoulli, 23(2), 2017, 1233–1278

work page 2017
[8]

Iksanov, A

A. Iksanov, A. Marynych, M. Meiners, Asymptotics of random processes with immigration II: Convergence to stationarity,Bernoulli, 23(2), 2017, 1279–1298

work page 2017
[9]

Kilbas, H.M

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo,Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006

work page 2006
[10]

Kingman,Poisson Processes,Oxford Studies in Probability, Clarendon Press, 1993, 108 pp

J.F.C. Kingman,Poisson Processes,Oxford Studies in Probability, Clarendon Press, 1993, 108 pp

work page 1993
[11]

Kl¨ uppelberg, C

C. Kl¨ uppelberg, C. K¨ uhn, Fractional Brownian motion as a weak limit of Poisson shot noise processes - with applications to finance,Stochastic Processes and their Applications, 2004, 113, 333–351

work page 2004
[12]

Kl¨ uppelberg, T

C. Kl¨ uppelberg, T. Mikosch, Explosive Poisson shot noise processes with applications to risk reserves,Bernoulli, 1 (1/2), 1995, 125–47

work page 1995
[13]

Kl¨ uppelberg, T

C. Kl¨ uppelberg, T. Mikosch, A. Sch¨ arf, Regular variation in the mean and stable limits for Poisson shot noise.Bernoulli, 2003, 9 (3), 467–496. [14]NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.1.10 of 2023- 06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. ...

work page 2003
[14]

Pang, M.S

G. Pang, M.S. Taqqu, Non-stationary self-similar Gaussian processes as scaling limits of power- law shot noise processes and generalizations of fractional Brownian motion,High Frequency, 2019, 2 (2), 95-112

work page 2019
[15]

G. Pang, Y. Zhou, Functional limit theorems for a new class of non-stationary shot noise processes,Stochastic Processes and their Applications, 128 (2), 2018, 505-544,

work page 2018
[16]

Schmidt, Shot-noise processes in finance, In: Ferger, D., Gonz´ alez Manteiga, W., Schmidt, T., Wang, JL

T. Schmidt, Shot-noise processes in finance, In: Ferger, D., Gonz´ alez Manteiga, W., Schmidt, T., Wang, JL. (eds) From Statistics to Mathematical Finance. Springer, Cham., 2017

work page 2017
[17]

Schottky, ¨Uber spontane Stromschwankungen in verschiedenen Elektrizit¨ atsleitern,An- nalen der Physik, 1918, 362, 541–567

W. Schottky, ¨Uber spontane Stromschwankungen in verschiedenen Elektrizit¨ atsleitern,An- nalen der Physik, 1918, 362, 541–567

work page 1918
[18]

Volterra, Sopra le funzioni che dipendono da altre funzioni

V. Volterra, Sopra le funzioni che dipendono da altre funzioni. Nota I,Rend. Lincei, Serie IV, 1887, 3, 97–105

work page
[19]

Volterra, Sopra le funzioni che dipendono da altre funzioni

V. Volterra, Sopra le funzioni che dipendono da altre funzioni. Nota II,Rend. Lincei, Serie IV, 1887, 3, 141–146

work page
[20]

Volterra, Sopra le funzioni che dipendono da altre funzioni

V. Volterra, Sopra le funzioni che dipendono da altre funzioni. Nota III,Rend. Lincei, Serie IV, 1887, 3, 153–158

work page
[21]

R. Wang, Y. Xiao, Q. Yang, Sample path moderate deviations for non-stationary power-law shot noise processes,Journal of Applied Probability, 2025, 1-27, Published online. SHOT-NOISE PROCESSES WITH LOGARITHMIC RESPONSE FUNCTION AND THEIR SCALING LIMITS 16 1 Department of Statistical Sciences, Sapienza University of Rome Email address:luisa.beghin@uniroma1....

work page 2025

[1] [1]

Beghin, L

L. Beghin, L. Cristofaro, F. Polito, Stochastic processes related to Hadamard operators and Le Roy measures,Advances in Differential Equations, 2026, 31 (1/2), 83-124

work page 2026

[2] [2]

Beghin, A

L. Beghin, A. De Gregorio, Y. Mishura, Hadamard fractional Brownian motion: path proper- ties and Wiener integration, 2025, arXiv:2507.13512v1

work page arXiv 2025

[3] [3]

P.L.Butzer, A.A.Kilbas, J.J.Trujillo, Fractional calculus in the Mellin setting and Hadamard- type fractional integrals,J. Math. Anal. Appl.,2002, 269, 1-27. SHOT-NOISE PROCESSES WITH LOGARITHMIC RESPONSE FUNCTION AND THEIR SCALING LIMITS 15

work page 2002

[4] [4]

Dassios, J

A. Dassios, J. Jang, Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity,Finance and Stochastics, 2003, 7(1), 73-95

work page 2003

[5] [5]

Dechant, E

A. Dechant, E. Lutz, Wiener-Khinchin theorem for nonstationary scale-invariant processes, Physical Rev. Lett., 2015, 115, 080603, 1-6

work page 2015

[6] [6]

Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions,Stochastic Processes and their Applications, 123(6), 2013, 1987-2010

A. Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions,Stochastic Processes and their Applications, 123(6), 2013, 1987-2010

work page 2013

[7] [7]

Iksanov, A

A. Iksanov, A. Marynych, M. Meiners, Asymptotics of random processes with immigration I: Scaling limits,Bernoulli, 23(2), 2017, 1233–1278

work page 2017

[8] [8]

Iksanov, A

A. Iksanov, A. Marynych, M. Meiners, Asymptotics of random processes with immigration II: Convergence to stationarity,Bernoulli, 23(2), 2017, 1279–1298

work page 2017

[9] [9]

Kilbas, H.M

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo,Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006

work page 2006

[10] [10]

Kingman,Poisson Processes,Oxford Studies in Probability, Clarendon Press, 1993, 108 pp

J.F.C. Kingman,Poisson Processes,Oxford Studies in Probability, Clarendon Press, 1993, 108 pp

work page 1993

[11] [11]

Kl¨ uppelberg, C

C. Kl¨ uppelberg, C. K¨ uhn, Fractional Brownian motion as a weak limit of Poisson shot noise processes - with applications to finance,Stochastic Processes and their Applications, 2004, 113, 333–351

work page 2004

[12] [12]

Kl¨ uppelberg, T

C. Kl¨ uppelberg, T. Mikosch, Explosive Poisson shot noise processes with applications to risk reserves,Bernoulli, 1 (1/2), 1995, 125–47

work page 1995

[13] [13]

Kl¨ uppelberg, T

C. Kl¨ uppelberg, T. Mikosch, A. Sch¨ arf, Regular variation in the mean and stable limits for Poisson shot noise.Bernoulli, 2003, 9 (3), 467–496. [14]NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.1.10 of 2023- 06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. ...

work page 2003

[14] [14]

Pang, M.S

G. Pang, M.S. Taqqu, Non-stationary self-similar Gaussian processes as scaling limits of power- law shot noise processes and generalizations of fractional Brownian motion,High Frequency, 2019, 2 (2), 95-112

work page 2019

[15] [15]

G. Pang, Y. Zhou, Functional limit theorems for a new class of non-stationary shot noise processes,Stochastic Processes and their Applications, 128 (2), 2018, 505-544,

work page 2018

[16] [16]

Schmidt, Shot-noise processes in finance, In: Ferger, D., Gonz´ alez Manteiga, W., Schmidt, T., Wang, JL

T. Schmidt, Shot-noise processes in finance, In: Ferger, D., Gonz´ alez Manteiga, W., Schmidt, T., Wang, JL. (eds) From Statistics to Mathematical Finance. Springer, Cham., 2017

work page 2017

[17] [17]

Schottky, ¨Uber spontane Stromschwankungen in verschiedenen Elektrizit¨ atsleitern,An- nalen der Physik, 1918, 362, 541–567

W. Schottky, ¨Uber spontane Stromschwankungen in verschiedenen Elektrizit¨ atsleitern,An- nalen der Physik, 1918, 362, 541–567

work page 1918

[18] [18]

Volterra, Sopra le funzioni che dipendono da altre funzioni

V. Volterra, Sopra le funzioni che dipendono da altre funzioni. Nota I,Rend. Lincei, Serie IV, 1887, 3, 97–105

work page

[19] [19]

Volterra, Sopra le funzioni che dipendono da altre funzioni

V. Volterra, Sopra le funzioni che dipendono da altre funzioni. Nota II,Rend. Lincei, Serie IV, 1887, 3, 141–146

work page

[20] [20]

Volterra, Sopra le funzioni che dipendono da altre funzioni

V. Volterra, Sopra le funzioni che dipendono da altre funzioni. Nota III,Rend. Lincei, Serie IV, 1887, 3, 153–158

work page

[21] [21]

R. Wang, Y. Xiao, Q. Yang, Sample path moderate deviations for non-stationary power-law shot noise processes,Journal of Applied Probability, 2025, 1-27, Published online. SHOT-NOISE PROCESSES WITH LOGARITHMIC RESPONSE FUNCTION AND THEIR SCALING LIMITS 16 1 Department of Statistical Sciences, Sapienza University of Rome Email address:luisa.beghin@uniroma1....

work page 2025