Almost periodicity as a path property for $p$-adic self-similar processes with stationary increments

Yi Shen; Zhenyuan Zhang

arxiv: 2605.17155 · v1 · pith:QQA67CNQnew · submitted 2026-05-16 · 🧮 math.PR

Almost periodicity as a path property for p-adic self-similar processes with stationary increments

Yi Shen , Zhenyuan Zhang This is my paper

Pith reviewed 2026-05-20 14:18 UTC · model grok-4.3

classification 🧮 math.PR

keywords p-adic self-similar processesstationary incrementsBohr almost periodicityp-adic continuityWeyl almost periodicityBesicovitch almost periodicityrandom fields

0 comments

The pith

For p-adic self-similar processes with stationary increments, Bohr almost periodicity of paths is equivalent to p-adic continuity.

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

The paper shows that in Banach space-valued p-adic self-similar processes with stationary increments, a sample path is Bohr almost periodic exactly when it is continuous with respect to the p-adic topology. This link does not hold for the Weyl or Besicovitch versions of almost periodicity. The authors also extend the Bohr equivalence to finite-dimensional random fields. A reader would care because almost periodicity often signals regular, repetitive behavior in stochastic processes, and tying it to a simple topological property like p-adic continuity could make checking this property easier in non-standard number systems.

Core claim

Bohr almost periodicity is equivalent, as a path event, to continuity with respect to the p-adic topology for these processes. The equivalence fails for Weyl and Besicovitch almost periodicity. The Bohr result extends to finite-dimensional random fields.

What carries the argument

The equivalence between Bohr almost periodicity and p-adic continuity as events on the sample paths of the p-adic sssi processes.

If this is right

Bohr almost periodic paths are precisely the p-adic continuous ones.
Weyl almost periodicity is not equivalent to p-adic continuity.
Besicovitch almost periodicity is not equivalent to p-adic continuity.
The equivalence for Bohr holds when extending to finite-dimensional random fields.

Where Pith is reading between the lines

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

This suggests that p-adic continuity might serve as a practical test for detecting almost periodic behavior in simulations of such processes.
The result could connect to studies of regularity in other non-Archimedean valued processes.
Extensions to infinite-dimensional cases might follow similar lines if the Banach space structure is preserved.

Load-bearing premise

The processes are Banach space-valued p-adic self-similar processes with stationary increments to which the standard definitions of almost periodicity apply directly on their sample paths.

What would settle it

A specific sample path from such a process that is continuous in the p-adic topology but fails to be Bohr almost periodic, or the reverse.

Figures

Figures reproduced from arXiv: 2605.17155 by Yi Shen, Zhenyuan Zhang.

**Figure 1.** Figure 1: Deterministic implications among the almostperiodicity (a.p.) and continuity notions used in the paper, for fixed q ≥ 1. The reverse implications are false in general. We end this section with a simple example that shows that deterministically, limit periodicity is indeed weaker than p-adic continuity, hence all the other almost periodicities mentioned above are also weaker than p-adic continuity. Exampl… view at source ↗

read the original abstract

Shen and Zhang (2021) showed that almost periodicity naturally arises in the spectral representation of discrete-time $p$-adic self-similar processes with stationary increments. In this paper, we study several notions of almost periodicity as sample path properties of Banach space-valued $p$-adic sssi processes. We prove that Bohr almost periodicity is equivalent, as a path event, to continuity with respect to the $p$-adic topology. We also show that the corresponding equivalence fails for Weyl and Besicovitch almost periodicity. Finally, we extend the Bohr almost-periodic result to finite-dimensional random fields.

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

Bohr almost periodicity matches p-adic continuity as a path property here, but the link from distributional self-similarity to pathwise almost periods needs checking.

read the letter

The paper's key result is that for these Banach space-valued p-adic self-similar processes with stationary increments, a sample path is Bohr almost periodic if and only if it is continuous in the p-adic topology. They also demonstrate that the same equivalence does not hold for Weyl or Besicovitch almost periodicity, and they carry the Bohr result over to finite-dimensional random fields. This builds usefully on the 2021 spectral representation by making almost periodicity a verifiable path event rather than just a spectral feature. The contrast with the other almost periodicity notions is helpful for understanding what is special about the Bohr version in this setting. The random field extension keeps things focused and shows the idea travels. A potential soft spot is the one raised in the stress-test: self-similarity and stationary increments are distributional, so continuous paths in the support might not automatically be almost periodic without additional structure. The paper probably leverages the specific form of the processes from the spectral work to make the equivalence go through, but it would be good to see how they ensure the almost periods exist pathwise. The citations look appropriate and the argument does not appear circular. This work is for researchers already interested in p-adic stochastic processes and almost periodic functions on non-Archimedean groups. Someone working on path regularity in self-similar processes could pick up the equivalences and the counterexamples for the other notions. It is solid enough to merit peer review, as the claims are concrete and the extension adds something new. I would recommend sending it out for refereeing, with reviewers asked to look closely at the proof of the equivalence to confirm it handles the distributional nature properly.

Referee Report

1 major / 2 minor

Summary. The manuscript studies notions of almost periodicity as sample-path properties of Banach space-valued p-adic self-similar processes with stationary increments (sssi processes). It claims to prove that Bohr almost periodicity is equivalent, as a path event, to continuity with respect to the p-adic topology. The equivalence is shown to fail for Weyl and Besicovitch almost periodicity, and the Bohr result is extended to finite-dimensional random fields.

Significance. If the claimed equivalences are rigorously established, the work supplies a concrete link between topological continuity and Bohr almost periodicity for paths of p-adic sssi processes. This distinction among almost-periodicity notions may prove useful for regularity analysis in non-Archimedean stochastic settings.

major comments (1)

[main theorem (as stated in the abstract)] The central claim equates two path events, yet self-similarity X(at) ≃ a^H X(t) and stationary increments are distributional properties. It is unclear from the stated result how p-adic continuity of an arbitrary sample path forces the existence of relatively dense almost periods in the p-adic group without an additional pathwise argument. A concrete derivation showing that continuity alone, inside the class of realizations, implies the Bohr property is required to support the equivalence.

minor comments (2)

Clarify whether the processes are assumed to satisfy the scaling and increment properties pathwise or only in finite-dimensional distributions.
Specify the precise Banach-space setting and the topology used for the p-adic continuity statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the distinction between distributional properties and pathwise behavior in the central equivalence. We address the comment below and will strengthen the manuscript with an explicit derivation.

read point-by-point responses

Referee: [main theorem (as stated in the abstract)] The central claim equates two path events, yet self-similarity X(at) ≃ a^H X(t) and stationary increments are distributional properties. It is unclear from the stated result how p-adic continuity of an arbitrary sample path forces the existence of relatively dense almost periods in the p-adic group without an additional pathwise argument. A concrete derivation showing that continuity alone, inside the class of realizations, implies the Bohr property is required to support the equivalence.

Authors: We agree that the equivalence between the two path events requires a transparent pathwise justification, as self-similarity and stationary increments are initially distributional. In the construction of the processes (via the spectral representation extending Shen and Zhang), these relations hold almost surely as functional equations on the sample paths. For any realization satisfying the pathwise scaling X(at) = a^H X(t) together with stationary increments, p-adic continuity directly yields a relatively dense set of almost periods in the p-adic topology by iterative application of the scaling to control the modulus of continuity. We will add a dedicated lemma isolating this implication and expand the proof of the main theorem to include the explicit construction of the almost periods from continuity alone. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence proved as independent path-property statement

full rationale

The paper cites Shen and Zhang (2021) only for background on spectral representations of discrete-time processes. The central result—an equivalence between Bohr almost periodicity and p-adic continuity as path events—is presented as a new mathematical statement for Banach-valued sample paths of sssi processes. No step reduces the claimed equivalence to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the derivation remains self-contained against the stated assumptions on the processes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on standard definitions of p-adic self-similar processes with stationary increments and the classical notions of Bohr, Weyl, and Besicovitch almost periodicity; no free parameters, ad-hoc axioms, or invented entities are indicated.

axioms (2)

domain assumption Standard definitions of p-adic self-similar processes with stationary increments in Banach spaces hold.
Invoked throughout the abstract as the setting for the path properties.
domain assumption Classical definitions of Bohr, Weyl, and Besicovitch almost periodicity extend to the sample paths of these processes.
Used to state the equivalences and failures.

pith-pipeline@v0.9.0 · 5622 in / 1313 out tokens · 40785 ms · 2026-05-20T14:18:26.576904+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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G. Samorodnitsky and M. S. Taqqu.Stable Non-Gaussian Random Processes: Sto- chastic Models with Infinite Variance. Stochastic Modeling, Chapman & Hall, New York, 1994

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work page arXiv 2026

[1] [1]

Amerio and G

L. Amerio and G. Prouse.Almost-Periodic Functions and Functional Equations. Van Nostrand Reinhold, New York, 1971

work page 1971

[2] [2]

A. S. Besicovitch. On generalized almost periodic functions.Proceedings of the London Mathematical Society(2)25(1926), 495–512

work page 1926

[3] [3]

P. H. Bezandry and T. Diagana.Almost Periodic Stochastic Processes. Springer, New York, 2011

work page 2011

[4] [4]

S. Bochner. Abstrakte fastperiodische Funktionen.Acta Mathematica61(1933), 149– 184

work page 1933

[5] [5]

H. Bohr. Zur Theorie der fastperiodischen Funktionen. I. Eine Verallgemeinerung der Theorie der Fourierreihen.Acta Mathematica45(1925), 29–127

work page 1925

[6] [6]

Corduneanu.Almost Periodic Functions

C. Corduneanu.Almost Periodic Functions. 2nd English ed., Chelsea Publishing Co., New York, 1989

work page 1989

[7] [7]

Da Prato and C

G. Da Prato and C. Tudor. Periodic and almost periodic solutions for semilinear stochastic equations.Stochastic Analysis and Applications13(1995), no. 1, 13–33

work page 1995

[8] [8]

Embrechts and M

P. Embrechts and M. Maejima.Selfsimilar Processes. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2002

work page 2002

[9] [9]

Gefferth, D

A. Gefferth, D. Veitch, I. Maricza, S. Moln´ ar, and I. Ruzsa. The nature of discrete second-order self-similarity.Advances in Applied Probability35(2003), no. 2, 395– 416

work page 2003

[10] [10]

Andres, A

J. Andres, A. M. Bersani, and R. F. Grande. Hierarchy of almost-periodic function spaces.Rendiconti di Matematica e delle Sue Applicazioni26(2006), no. 2, 121–188

work page 2006

[11] [11]

T. Kawata. Almost periodic weakly stationary processes. InStatistics and Probability: Essays in Honor of C. R. Rao, G. Kallianpur, P. R. Krishnaiah, and J. K. Ghosh (eds.), pp. 383–396, North-Holland, Amsterdam-New York, 1982

work page 1982

[12] [12]

Kˆ ono and M

N. Kˆ ono and M. Maejima. Self-similar stable processes with stationary increments. InStable Processes and Related Topics, S. Cambanis, G. Samorodnitsky and M. S. Taqqu (eds.), pp. 275–295, Birkh¨ auser, Boston, 1991

work page 1991

[13] [13]

Lamperti

J. Lamperti. Semi-stable stochastic processes.Transactions of the American Mathe- matical Society104(1962), 62–78

work page 1962

[14] [14]

B. M. Levitan and V. V. Zhikov.Almost Periodic Functions and Differential Equa- tions. Cambridge University Press, Cambridge, 1982

work page 1982

[15] [15]

Pipiras and M

V. Pipiras and M. S. Taqqu.Long-Range Dependence and Self-Similarity. Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2017. ALMOST PERIODICITY AS A PATH PROPERTY FORP-ADIC SSSI PROCESSES 15

work page 2017

[16] [16]

Samorodnitsky and M

G. Samorodnitsky and M. S. Taqqu.Stable Non-Gaussian Random Processes: Sto- chastic Models with Infinite Variance. Stochastic Modeling, Chapman & Hall, New York, 1994

work page 1994

[17] [17]

Shen and Z

Y. Shen and Z. Zhang. On discrete-time self-similar processes with stationary incre- ments.Electronic Journal of Probability26(2021), 1–24

work page 2021

[18] [18]

E. Slutsky. Sur les fonctions al´ eatoires presque p´ eriodiques et sur la d´ ecomposition des fonctions al´ eatoires stationnaires en composantes.Actualit´ es scientifiques et in- dustrielles, no. 738, Hermann, Paris, 1938, pp. 33–55

work page 1938

[19] [19]

M. Udagawa. Asymptotic properties of distributions of some functionals of random variables.Reports of Statistical Application Research, Union of Japanese Scientists and Engineers2(1952), nos. 2–3, 1–66

work page 1952

[20] [20]

W. Vervaat. Sample path properties of self-similar processes with stationary incre- ments.The Annals of Probability13(1985), no. 1, 1–27

work page 1985

[21] [21]

H. Weyl. Integralgleichungen und fastperiodische Funktionen.Mathematische An- nalen97(1927), 338–356

work page 1927

[22] [22]

Z. Zhang. Boundedness of discounted branching random walks via generic chaining. arXiv preprint arXiv:2602.18919, 2026

work page arXiv 2026