Persistence probabilities for fractionally integrated fractional Brownian noise

G. Molchan

arxiv: 2509.10265 · v10 · pith:A6S4ZT5Xnew · submitted 2025-09-12 · 🧮 math.PR

Persistence probabilities for fractionally integrated fractional Brownian noise

G. Molchan This is my paper

Pith reviewed 2026-05-21 23:00 UTC · model grok-4.3

classification 🧮 math.PR

keywords persistence exponentfractional Brownian motionfractional integrationSlepian lemmaGaussian processespersistence probabilitylong-range dependence

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The pith

The persistence exponent e(a,H) for fractionally integrated fractional Brownian noise equals e(a + 2H - 1, 1 - H).

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

This paper examines the persistence probability for a process obtained by fractionally integrating fractional Brownian motion, parameterized by integration multiplicity a and Hurst index H. It focuses on the exponent e(a,H) that governs the power-law decay of the probability that the process remains below a fixed level over long times (0,T). The central result is an identity showing that this exponent is the same for the parameter pairs (a,H) and (a+2H-1,1-H). This identity is used to refute the long-held conjecture that the exponent equals H(1-H) when a=2, and it also yields the exact exponent for large a. Sympathetic readers would care because these exponents control the likelihood of long excursions in processes with long-range dependence, which appear in physics, finance, and hydrology.

Core claim

For the fractionally integrated fractional Brownian noise I(t; a, H) in the region where a + H > 1 and 0 < H < 1, the persistence exponent satisfies e(a, H) = e(a + 2H - 1, 1 - H). This identity refutes the hypothesis that e(2, H) = H(1 - H) and provides the asymptotic value of e(a, H) for large a as the persistence exponent of a specific stationary Gaussian process with covariance cosh((H - 1/2)t)/cosh(t/2).

What carries the argument

A generalization of Slepian's lemma applicable to families of Gaussian processes that depend smoothly on a continuous parameter, combined with a continuity lemma for persistence exponents.

If this is right

The exponent decreases as the integration multiplicity a increases.
Near the boundary of the domain G, the exponents exhibit specific limiting behaviors, including at infinity.
Fractional Brownian motion with parameter H is related to that with 1-H through a fractional integration operation.
Exact values are obtained for e(a, H) when a is much larger than 1.

Where Pith is reading between the lines

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

This symmetry might extend to other integrated Gaussian processes beyond fractional Brownian motion.
Numerical simulations of persistence probabilities for specific values like a=2 and H=0.7 could independently verify the refutation of the old hypothesis.
The relation between FBM(H) and FBM(1-H) suggests a duality in long-memory behaviors under integration.

Load-bearing premise

The generalized Slepian lemma applies to the family of fractionally integrated processes that depend smoothly on the multiplicity parameter a inside the region G = (a + H > 1, 0 < H < 1).

What would settle it

A direct computation or simulation of the long-time decay rate of the persistence probability for parameters (a, H) and (a + 2H - 1, 1 - H) showing that the two rates differ.

read the original abstract

The main objective of this study is fractionally integrated fractional Brownian noise, I(t/a,H) where a>0 is the 'multiplicity' of integration, and H is the Hurst parameter . The subject of the analysis is the persistence exponent e(a,H) that determines the power-law asymptotic of probability that the process will not exceed a fixit level in a growing time interval (0,T). In the important cases such as fractional Brownian motion(FBM(H),a=1) and integtated Wienr process(a=2,H=1/2) these exponents are well known. To understand the problematic exponents e(2,H), we consider the (a,H) parameters from the maximum (for the task) area G= (a+H>1,0<H<1) ). We prove the decrease of the exponents with increasing 'a' and describe their behavior near the boundary of G, including infinity. The identity of the exponents with parameters (a,H) and (a+2H-1,1-H) has been established. On this way, the long-standing hypothesis that e(2,H)=H(1-H) has been refuted. In addition, it has been revealed that FBM(H) and FBM(1-H) processes are related by a fractional integration operation. We have obtained the exact value of the exponent for e(a>>1, H). It is identical to the persistence exponent for a Gaussian stationary process with covariance cosh((H-1/2)t)/cosh(t/2) and generalizes the well-known case of H=1/2. Our results use well known the continuity lemma for the persistence exponents and a some generalization of Slepian's lemma for a family of Gaussian processes smoothly dependent on a parameter.

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 proves a symmetry e(a,H)=e(a+2H-1,1-H) for persistence exponents and refutes the e(2,H)=H(1-H) conjecture, but the generalized Slepian application rests on an unverified smoothness claim for the covariance in a.

read the letter

The main things to know are the symmetry identity connecting the persistence exponents at (a,H) and (a+2H-1,1-H), plus the explicit refutation of the long-standing guess that e(2,H) equals H(1-H). The paper also supplies an exact large-a formula that matches the known stationary Gaussian case when H=1/2 and describes how the exponents behave near the boundary of the region G and decrease with a. These are concrete advances over the abstract-only sketch of known cases like plain fBM and integrated Wiener process. The work ties FBM(H) and FBM(1-H) together via fractional integration, which is a clean observation. It leans on the standard continuity lemma for exponents and a generalized Slepian comparison for parameter-dependent families, both of which are reasonable tools when the hypotheses hold. The large-a result looks like a direct and useful extension. The soft spot is exactly the one the stress-test flags: the generalized Slepian lemma needs the covariance of the integrated process to be C^1 in the multiplicity a, uniformly on compact sets inside G. The abstract does not show this differentiability explicitly, especially as a approaches the boundary a+H=1 or when H varies. If the full text supplies a direct check that the fractional integral kernel satisfies the required smoothness without extra restrictions, the symmetry and refutation go through; otherwise the comparison between the two parameter pairs is not justified. No circularity or fitting appears in the claims. This is for specialists in persistence probabilities and long-memory Gaussian processes who already know the continuity and Slepian tools. A reader looking for exact relations or counterexamples to old conjectures will find value. The paper deserves a serious referee so the lemma application can be verified in detail. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the persistence exponent e(a,H) for the fractionally integrated fractional Brownian noise I(t;a,H) in the region G=(a+H>1, 0<H<1). It establishes that e(a,H) decreases with a, describes its behavior near the boundary of G and at infinity, proves the identity e(a,H)=e(a+2H-1,1-H), refutes the conjecture e(2,H)=H(1-H), and obtains the exact large-a asymptotic, which coincides with the persistence exponent of the stationary Gaussian process having covariance cosh((H-1/2)t)/cosh(t/2). The arguments rely on a continuity lemma for persistence exponents and a generalized Slepian lemma for one-parameter families of Gaussian processes whose covariance depends smoothly on a.

Significance. If the central claims are established, the work resolves a long-standing open question on the persistence exponent for integrated fractional Brownian motion and supplies a useful symmetry that relates exponents across different integration multiplicities and Hurst parameters. The large-a exact result is a concrete advance that generalizes the classical H=1/2 stationary case. The technique of applying a parameter-dependent Slepian comparison, once the smoothness hypothesis is verified, could be reusable for other families of Gaussian processes.

major comments (2)

[Application of the generalized Slepian lemma (proof of the symmetry identity)] The proof of the identity e(a,H)=e(a+2H-1,1-H) invokes the generalized Slepian lemma on the one-parameter family of processes I(t;a,H). The lemma requires that the covariance be C^1 in the multiplicity parameter a, uniformly on compact subsets of G. The manuscript constructs the covariance via a fractional integral kernel but does not supply the differentiation under the integral sign or the uniform estimates needed to confirm this C^1 dependence, particularly as a+H approaches 1 from above. Without this verification the comparison between the two parameter pairs does not follow and the refutation of e(2,H)=H(1-H) remains conditional.
[Continuity argument following the Slepian comparison] The continuity lemma for persistence exponents is cited to pass from the Slepian comparison to the equality of the exponents. The manuscript does not record the precise hypotheses of this continuity lemma (e.g., uniform integrability or modulus-of-continuity requirements on the covariance) nor verify that they hold for the family I(t;a,H) inside G. This step is load-bearing for all quantitative statements about e(a,H).

minor comments (2)

[Abstract] Abstract contains several typographical and grammatical errors: 'fixit level' should read 'fixed level'; 'integtated Wienr process' should read 'integrated Wiener process'; the clause 'Our results use well known the continuity lemma...' is ungrammatical and should be rephrased.
[Introduction and notation section] Notation for the process is introduced as I(t/a,H) in the abstract but later appears as I(t;a,H). A single consistent notation should be adopted throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment below and will make the necessary revisions to strengthen the presentation of our results.

read point-by-point responses

Referee: [Application of the generalized Slepian lemma (proof of the symmetry identity)] The proof of the identity e(a,H)=e(a+2H-1,1-H) invokes the generalized Slepian lemma on the one-parameter family of processes I(t;a,H). The lemma requires that the covariance be C^1 in the multiplicity parameter a, uniformly on compact subsets of G. The manuscript constructs the covariance via a fractional integral kernel but does not supply the differentiation under the integral sign or the uniform estimates needed to confirm this C^1 dependence, particularly as a+H approaches 1 from above. Without this verification the comparison between the two parameter pairs does not follow and the refutation of e(2,H)=H(1-H) remains conditional.

Authors: We agree that explicit verification of the C^1 regularity in the parameter a is essential for applying the generalized Slepian lemma. In the revised manuscript, we will add a new lemma or proposition that establishes the required differentiability. Specifically, we will differentiate the covariance expression under the integral sign, justifying this step via dominated convergence or similar arguments using the properties of the fractional integral kernel. We will also provide uniform estimates on compact subsets of G, including as a + H approaches 1 from above, by leveraging bounds on the kernel and its derivatives. This will ensure the applicability of the lemma and solidify the symmetry identity and the refutation of the conjecture. revision: yes
Referee: [Continuity argument following the Slepian comparison] The continuity lemma for persistence exponents is cited to pass from the Slepian comparison to the equality of the exponents. The manuscript does not record the precise hypotheses of this continuity lemma (e.g., uniform integrability or modulus-of-continuity requirements on the covariance) nor verify that they hold for the family I(t;a,H) inside G. This step is load-bearing for all quantitative statements about e(a,H).

Authors: We acknowledge the need to make the application of the continuity lemma fully rigorous. In the revision, we will first state the precise hypotheses of the continuity lemma as used in the literature. Subsequently, we will verify that these hypotheses are satisfied by the family of processes I(t; a, H) for (a, H) in G. This includes checking uniform integrability of the relevant quantities and establishing a suitable modulus of continuity for the covariance functions that is uniform in the parameter on compact sets. These additions will clarify the justification for equating the persistence exponents after the Slepian comparison. revision: yes

Circularity Check

0 steps flagged

No circularity: identities derived from external lemmas on independent covariance properties

full rationale

The paper establishes the key identity e(a,H)=e(a+2H-1,1-H) and refutes e(2,H)=H(1-H) by applying a generalized Slepian lemma and a continuity lemma to the covariance structure of the fractionally integrated process I(t;a,H) inside the open region G. These lemmas are presented as known or generalized tools whose statements depend only on smoothness of the covariance in the multiplicity parameter a, not on the target exponent values themselves. No equation or step redefines an output as an input, renames a fitted quantity as a prediction, or closes a self-citation loop that would force the result by construction. The derivation remains self-contained against the stated external lemmas.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on two standard tools from Gaussian process theory whose applicability to the integrated fractional noise family is asserted but not derived in the abstract.

axioms (2)

standard math Continuity lemma for persistence exponents
Invoked to establish monotonicity and boundary behavior of e(a,H).
standard math Generalized Slepian lemma for smoothly parameter-dependent Gaussian processes
Used to compare processes and obtain the symmetry and large-a limits.

pith-pipeline@v0.9.0 · 5846 in / 1441 out tokens · 43032 ms · 2026-05-21T23:00:30.757442+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 dAlembert_cosh_solution_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

limiting correlation function ... cosh((H-1/2)t)/cosh(t/2) ... persistence exponent ... Statement 1.4
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean cost_alpha_one_eq_jcost echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

spectral symmetry ... (a,H) and (a+2H-1,1-H) ... equality of exponents

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

16 extracted references · 16 canonical work pages

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and Simon T

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Aurzada F., Kilian M. Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion .Probability Theory and their Applications, 67(1), 100-114 (2022), https://www.mathnet.ru/tvp5423 19

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and Mukherjee S

Aurzada F. and Mukherjee S. Persistence probabilities of weighted sums of stationary Gaussian sequences. Stochastic Processes and their Applications, 159:286–319 (2023)

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Bateman G., Erdelyi A. Higher transcendental functions, v.1, NY,Toronto, London, Mc Graw-Hill book Company, Inc. (1953). 6.Bray A., Majumdar S., Schehr G. Persistence and First-Passage Properties in Non- equilibrium System. Advances in Physics, 62(3), 225-361 (2013). 7.Dembo A. and Mukherjee.S. No zero-crossings for random polynomials and the heat equatio...

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Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

Ruzmaikina A. Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion. J. Stat. Phys. 100, 1049–1069 (2000). 19.She, Z., Aurell, E., Frish, U. The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–642 (1992)

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Distribution of some functionals of the integrals of Brownian motion

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Bell System Tech

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[1] [1]

and Dereich S

AurzadaF. and Dereich S. Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann.Inst.H.Poincare Probab. Stat. 49(1), 236-251(2013)

work page 2013

[2] [2]

and Simon T

Aurzada F. and Simon T. Persistence probabilities and exponents. Lecture Notes in Math., 2149 ,183–224. Springer Verlag (2015)

work page 2015

[3] [3]

Aurzada F., Kilian M. Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion .Probability Theory and their Applications, 67(1), 100-114 (2022), https://www.mathnet.ru/tvp5423 19

work page 2022

[4] [4]

and Mukherjee S

Aurzada F. and Mukherjee S. Persistence probabilities of weighted sums of stationary Gaussian sequences. Stochastic Processes and their Applications, 159:286–319 (2023)

work page 2023

[5] [5]

Higher transcendental functions, v.1, NY,Toronto, London, Mc Graw-Hill book Company, Inc

Bateman G., Erdelyi A. Higher transcendental functions, v.1, NY,Toronto, London, Mc Graw-Hill book Company, Inc. (1953). 6.Bray A., Majumdar S., Schehr G. Persistence and First-Passage Properties in Non- equilibrium System. Advances in Physics, 62(3), 225-361 (2013). 7.Dembo A. and Mukherjee.S. No zero-crossings for random polynomials and the heat equatio...

work page 1953

[6] [6]

Persistence and Ball Exponents for Gaussian Stationary Processes

Feldheim, N.D., Feldheim, O.N ,Mukherjee S. Persistence and Ball Exponents for Gaussian Stationary Processes. Preprint. arXiv:2112.04820 (2021) 9.Feldheim, N.D., Feldheim, O.N., Nitzan, S. Persistence of Gaussian stationary processes: a spectral perspective. Ann. Probab. 49(3), 1067–1096 (2021):

work page arXiv 2021

[7] [7]

Lectures on Gaussian Processes.246 p, Springer, New York (2012) .11

Lifshits, M. Lectures on Gaussian Processes.246 p, Springer, New York (2012) .11. Molchan, G. Maximum of fractional Brownian motion: probabilities of small values. Commun. Math. Phys. 205(1), 97–11 (1999)

work page 2012

[8] [8]

Small values of the maximum for the integral of fractional Brownian motion , J

Molchan G., Khokhlov A. Small values of the maximum for the integral of fractional Brownian motion , J. Stat. Phys., 114(3-4), 923-946 (2004)

work page 2004

[9] [9]

Survival Exponents for Some Gaussian Processes

Molchan G. Survival Exponents for Some Gaussian Processes. International Journal of Stochastic Analysis, 2012, Article ID 137271, Hindawi Publishing Corporation (2012) , doi:10.1155/2012/37271

work page doi:10.1155/2012/37271 2012

[10] [10]

The Inviscid Burgers Equation with Fractional Brownian Initial Data: The Dimension of Regular Lagrangian points

Molchan G. The Inviscid Burgers Equation with Fractional Brownian Initial Data: The Dimension of Regular Lagrangian points. J. Stat. Phys.,. 167(6), 1546–1554 (2017)

work page 2017

[11] [11]

Pitman, E. J. G. On the behavior of the characteristic function of a probability distribution in the neighborhood of the origin, J. Austral. Math. Soc. 8 , 422–443 (1968)

work page 1968

[12] [12]

Persistence of integrated stable processes

Profeta Ch., Simon T. Persistence of integrated stable processes. Probability theory and related fields, 162,463-485 (2015). 20

work page 2015

[13] [13]

Persistence exponent for the 2d-diffusion equation and related Kac polynomials, Phys

Poplavskyi M., Schehr G. Persistence exponent for the 2d-diffusion equation and related Kac polynomials, Phys. Rev. Lett. 121, 150601 (2018)

work page 2018

[14] [14]

Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

Ruzmaikina A. Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion. J. Stat. Phys. 100, 1049–1069 (2000). 19.She, Z., Aurell, E., Frish, U. The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–642 (1992)

work page 2000

[15] [15]

Distribution of some functionals of the integrals of Brownian motion

Sinai, Ya.G. Distribution of some functionals of the integrals of Brownian motion. Theor. Math. Phys. 90(3), 219-241 (1992)

work page 1992

[16] [16]

Bell System Tech

Slepian D.(1962) The one-sided barrier problem for Gaussian noise. Bell System Tech. J., 41:463–501 (1962)

work page 1962