pith. sign in

arxiv: 2605.23659 · v1 · pith:KNA53QVLnew · submitted 2026-05-22 · 🧮 math.PR

State-dependent inverse-subordinator time changes of regenerative processes: Excursion structure and multiscale occupation-time limits

Kosuke Yamato This is my paper

Pith reviewed 2026-05-25 03:42 UTC · model grok-4.3

classification 🧮 math.PR
keywords regenerative processesinverse subordinatorsexcursion theoryoccupation timesregular variationPoisson point processesarcsine lawsDarling-Kac limits
0
0 comments X

The pith

State-dependent inverse-subordinator time changes preserve a Poisson excursion point process in regenerative processes.

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

The paper studies regenerative processes modified by state-dependent inverse subordinators that assign independent subordinators to different measurable classes of excursions and construct a random clock from the resulting occupation times. It shows that the time-changed process is generally not regenerative yet its transformed excursion point process remains Poisson through an excursion-wise marking-and-mapping procedure. This structure is then used to derive a multiscale joint occupation-time limit theorem when the transformed excursion lifetime tails are regularly varying, yielding generalized arcsine laws and Darling-Kac type limits. A sympathetic reader would care because the result extends classical occupation-time asymptotics to flexible state-dependent clocks where ordinary regenerativity no longer holds.

Core claim

Although the resulting process is generally not regenerative, we prove that its transformed excursion point process is again Poisson, using an excursion-wise marking-and-mapping procedure. Under regular variation assumptions on the transformed excursion lifetime tails, we prove a multiscale joint occupation-time limit theorem, including generalized arcsine laws and Darling-Kac type limits.

What carries the argument

Excursion-wise marking-and-mapping procedure that converts the state-dependent time-changed excursions into a Poisson point process.

If this is right

  • Generalized arcsine laws describe the occupation-time distribution of the time-changed process.
  • Darling-Kac type limits apply to occupation times at multiple scales.
  • Joint multiscale limits exist for the vector of occupation times across different scaling regimes.
  • The Poisson property of the transformed excursion process supplies the necessary independence for these asymptotic results.

Where Pith is reading between the lines

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

  • The marking-and-mapping technique could be tested on non-regenerative processes that still possess an excursion decomposition.
  • Relaxing regular variation might produce different scaling regimes or stable-law limits instead of the multiscale arcsine family.
  • The construction suggests direct modeling of state-dependent clocks in applied contexts such as storage or queueing systems.

Load-bearing premise

The tails of the transformed excursion lifetimes satisfy regular variation.

What would settle it

An explicit construction of a regenerative process and state-dependent subordinators where the transformed excursion point process fails to be Poisson, or where the regular-variation tail condition holds but the multiscale occupation-time limits deviate from the stated form.

read the original abstract

We study regenerative processes time-changed by state-dependent inverse subordinators. The construction assigns possibly different independent subordinators to measurable classes of excursions and builds a random clock from the corresponding occupation times. Although the resulting process is generally not regenerative, we prove that its transformed excursion point process is again Poisson, using an excursion-wise marking-and-mapping procedure. We apply this to study occupation-time asymptotics. Under regular variation assumptions on the transformed excursion lifetime tails, we prove a multiscale joint occupation-time limit theorem, including generalized arcsine laws and Darling--Kac type limits.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript examines regenerative processes time-changed by state-dependent inverse subordinators, where different independent subordinators are assigned to measurable classes of excursions to construct a random clock from the corresponding occupation times. Although the resulting process is generally not regenerative, the authors prove that its transformed excursion point process remains Poisson via an excursion-wise marking-and-mapping procedure. Under regular variation assumptions on the transformed excursion lifetime tails, they establish a multiscale joint occupation-time limit theorem that includes generalized arcsine laws and Darling–Kac type limits.

Significance. If the central claims hold, the work extends classical excursion theory to state-dependent time changes while preserving the Poisson structure of the excursion point process. This provides a technically useful tool for deriving occupation-time asymptotics in non-regenerative settings. The explicit multiscale limits under regular variation assumptions strengthen the applicability to processes with heterogeneous scaling, and the marking-and-mapping technique appears to be a clean technical contribution.

major comments (2)
  1. [Abstract, §1] Abstract and §1: The Poisson property for the transformed excursion point process is stated as holding generally via the marking-and-mapping procedure, yet the multiscale occupation-time limits in the subsequent theorem require regular variation on the transformed excursion lifetime tails. The manuscript should clarify whether the Poisson preservation itself imposes any hidden tail conditions or whether the two results are fully decoupled; this distinction is load-bearing for the scope of the main theorems.
  2. [Abstract] The construction assigns subordinators to measurable classes of excursions. It is not immediately clear from the abstract how the measurability of the classes interacts with the state-dependence to guarantee the independence needed for the Poisson property; a concrete counter-example or boundary case when the classes are not sufficiently separated would strengthen the claim.
minor comments (2)
  1. [§2] Notation for the state-dependent inverse subordinators and the resulting random clock should be introduced with a single consistent symbol set early in §2 to avoid later ambiguity when referring to occupation times.
  2. [Abstract] The abstract mentions 'generalized arcsine laws and Darling–Kac type limits' as part of the multiscale theorem; a brief statement of the precise form of these limits (e.g., the normalizing sequences) would improve readability without lengthening the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: The Poisson property for the transformed excursion point process is stated as holding generally via the marking-and-mapping procedure, yet the multiscale occupation-time limits in the subsequent theorem require regular variation on the transformed excursion lifetime tails. The manuscript should clarify whether the Poisson preservation itself imposes any hidden tail conditions or whether the two results are fully decoupled; this distinction is load-bearing for the scope of the main theorems.

    Authors: The Poisson property is established in full generality by the marking-and-mapping procedure and imposes no tail conditions whatsoever. Regular variation is used exclusively for the multiscale occupation-time limit theorems. The two results are therefore fully decoupled; we will insert an explicit clarifying sentence in the abstract and in §1. revision: yes

  2. Referee: [Abstract] The construction assigns subordinators to measurable classes of excursions. It is not immediately clear from the abstract how the measurability of the classes interacts with the state-dependence to guarantee the independence needed for the Poisson property; a concrete counter-example or boundary case when the classes are not sufficiently separated would strengthen the claim.

    Authors: Measurability guarantees that the classes form a measurable partition of the excursion space, so that independent subordinators can be assigned without overlap. State-dependence is realized only through the occupation times inside each class; the independence of the subordinators is preserved by construction and is what the marking-and-mapping procedure exploits to retain the Poisson property. We will add a brief remark in the introduction that sketches a boundary case (non-measurable or overlapping classes) in which the required independence fails, thereby illustrating the necessity of the measurability assumption. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external assumptions

full rationale

The paper constructs state-dependent inverse-subordinator time changes on regenerative processes, proves the transformed excursion point process remains Poisson via an excursion-wise marking-and-mapping procedure, and derives multiscale occupation-time limits under explicitly stated regular-variation tail conditions on transformed excursion lifetimes. These tail conditions are external hypotheses (not fitted internally or derived from the result itself), the Poisson property is established directly from the construction without reducing to a self-citation or renaming, and no load-bearing step equates a prediction to its own input by definition. The central claims rest on standard excursion theory plus the stated assumptions rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the regular-variation tail assumptions for transformed excursion lifetimes and on background facts from excursion theory for regenerative processes. No free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Regular variation assumptions on the transformed excursion lifetime tails
    Explicitly invoked in the abstract as the condition under which the multiscale joint occupation-time limit theorem holds.

pith-pipeline@v0.9.0 · 5616 in / 1276 out tokens · 87610 ms · 2026-05-25T03:42:06.999470+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Baeumer and M

    B. Baeumer and M. M. Meerschaert. Stochastic solutions for fractional C auchy problems. Fract. Calc. Appl. Anal., 4 0 (4): 0 481--500, 2001. ISSN 1311-0454

    work page 2001

  2. [2]

    Barlow, J

    M. Barlow, J. Pitman, and M. Yor. Une extension multidimensionnelle de la loi de l'arc sinus. In S\'eminaire de P robabilit\'es, XXIII , volume 1372 of Lecture Notes in Math., pages 294--314. Springer, Berlin, 1989. ISBN 3-540-51191-1. doi:10.1007/BFb0083980. URL https://doi.org/10.1007/BFb0083980

    work page doi:10.1007/bfb0083980 1989

  3. [3]

    J. Bertoin. L\' e vy processes , volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996. ISBN 0-521-56243-0

    work page 1996

  4. [4]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1987. ISBN 0-521-30787-2. doi:10.1017/CBO9780511721434. URL https://doi.org/10.1017/CBO9780511721434

    work page doi:10.1017/cbo9780511721434 1987

  5. [5]

    R. M. Blumenthal and R. K. Getoor. Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29. Academic Press, New York-London, 1968

    work page 1968

  6. [6]

    Z.-Q. Chen. Time fractional equations and probabilistic representation. Chaos Solitons Fractals, 102: 0 168--174, 2017. ISSN 0960-0779,1873-2887. doi:10.1016/j.chaos.2017.04.029. URL https://doi.org/10.1016/j.chaos.2017.04.029

    work page doi:10.1016/j.chaos.2017.04.029 2017

  7. [7]

    Choi and D

    J. Choi and D. Clancy, Jr. Occupation times for time-changed processes with applications to parisian options, 2021. URL https://arxiv.org/abs/2110.07639

    work page arXiv 2021

  8. [8]

    Fujihara, Y

    E. Fujihara, Y. Kawamura, and Y. Yano. Functional limit theorems for occupation times of L amperti's stochastic processes in discrete time. J. Math. Kyoto Univ., 47 0 (2): 0 429--440, 2007. ISSN 0023-608X. doi:10.1215/kjm/1250281054. URL https://doi.org/10.1215/kjm/1250281054

    work page doi:10.1215/kjm/1250281054 2007

  9. [9]

    It\^o and H

    K. It\^o and H. P. McKean, Jr. Diffusion processes and their sample paths, volume Band 125 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin-New York, 1974. Second printing, corrected

    work page 1974

  10. [10]

    Jacod and A

    J. Jacod and A. N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2003. ISBN 3-540-43932-3. doi:10.1007/978-3-662-05265-5. URL https://doi.org/10.1007/978-3-662-05265-5

    work page doi:10.1007/978-3-662-05265-5 2003

  11. [11]

    Kallenberg

    O. Kallenberg. Foundations of modern probability, volume 99 of Probability Theory and Stochastic Modelling. Springer, Cham, Cham, 2021. ISBN 978-3-030-61871-1; 978-3-030-61870-4. doi:10.1007/978-3-030-61871-1. URL https://doi.org/10.1007/978-3-030-61871-1. Third edition [of 1464694]

    work page doi:10.1007/978-3-030-61871-1 2021

  12. [12]

    Kasahara

    Y. Kasahara. Spectral theory of generalized second order differential operators and its applications to M arkov processes. Japan. J. Math. (N.S.), 1 0 (1): 0 67--84, 1975/76. ISSN 0289-2316. doi:10.4099/math1924.1.67. URL https://doi.org/10.4099/math1924.1.67

    work page doi:10.4099/math1924.1.67 1975

  13. [13]

    Lamperti

    J. Lamperti. An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc., 88: 0 380--387, 1958. ISSN 0002-9947,1088-6850. doi:10.2307/1993222. URL https://doi.org/10.2307/1993222

    work page doi:10.2307/1993222 1958

  14. [14]

    Lamperti

    J. Lamperti. An invariance principle in renewal theory. Ann. Math. Statist., 33: 0 685--696, 1962. ISSN 0003-4851. doi:10.1214/aoms/1177704590. URL https://doi.org/10.1214/aoms/1177704590

    work page doi:10.1214/aoms/1177704590 1962

  15. [15]

    M. M. Meerschaert and A. Sikorskii. Stochastic models for fractional calculus, volume 43 of De Gruyter Studies in Mathematics. De Gruyter, Berlin, second edition, 2019. ISBN 978-3-11-055907-1; 978-3-11-056024-4; 978-3-11-055914-9. doi:10.1515/9783110560244. URL https://doi.org/10.1515/9783110560244

    work page doi:10.1515/9783110560244 2019

  16. [16]

    T. Sera. Functional limit theorem for occupation time processes of intermittent maps. Nonlinearity, 33 0 (3): 0 1183--1217, 2020. ISSN 0951-7715,1361-6544. doi:10.1088/1361-6544/ab5ceb. URL https://doi.org/10.1088/1361-6544/ab5ceb

    work page doi:10.1088/1361-6544/ab5ceb 2020

  17. [17]

    C. Stone. Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Amer. Math. Soc., 14: 0 694--696, 1963. ISSN 0002-9939,1088-6826. doi:10.2307/2034973. URL https://doi.org/10.2307/2034973

    work page doi:10.2307/2034973 1963

  18. [18]

    B. Toaldo. Convolution-type derivatives, hitting-times of subordinators and time-changed C_0 -semigroups. Potential Anal., 42 0 (1): 0 115--140, 2015. ISSN 0926-2601,1572-929X. doi:10.1007/s11118-014-9426-5. URL https://doi.org/10.1007/s11118-014-9426-5

    work page doi:10.1007/s11118-014-9426-5 2015

  19. [19]

    Watanabe

    S. Watanabe. Generalized arc-sine laws for one-dimensional diffusion processes and random walks. In Stochastic analysis ( I thaca, NY , 1993) , volume 57 of Proc. Sympos. Pure Math., pages 157--172. Amer. Math. Soc., Providence, RI, 1995. ISBN 0-8218-0289-5. doi:10.1090/pspum/057/1335470. URL https://doi.org/10.1090/pspum/057/1335470

    work page doi:10.1090/pspum/057/1335470 1993

  20. [20]

    Y. Yano. On the joint law of the occupation times for a diffusion process on multiray. J. Theoret. Probab., 30 0 (2): 0 490--509, 2017. ISSN 0894-9840,1572-9230. doi:10.1007/s10959-015-0654-4. URL https://doi.org/10.1007/s10959-015-0654-4

    work page doi:10.1007/s10959-015-0654-4 2017