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arxiv: 2605.21005 · v1 · pith:SBEC4SQSnew · submitted 2026-05-20 · 🧮 math.PR · math-ph· math.MP

Double-transform Tauberian method for precise large deviations

Giampaolo Cristadoro , Gaia Pozzoli This is my paper

Pith reviewed 2026-05-21 02:19 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords large deviationsTauberian theoremsstable lawsstochastic processesbivariate transformsrandom walksprecise asymptotics
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The pith

The double-transform Tauberian method provides precise large deviation asymptotics for bivariate stochastic processes in the domain of spectrally positive stable laws.

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

This paper extends a Tauberian approach, previously used for single-variable Laplace-Stieltjes transforms, to handle double transforms that couple space and time variables. The goal is to derive precise large deviations for stochastic processes whose distributions are attracted to spectrally positive stable laws. Such extensions are useful because many models encode observables in bivariate transforms that are analytically tractable even when increments are correlated or paths are constrained. The method offers a direct way to characterize asymptotic behavior that single-transform techniques struggle with.

Core claim

By developing a bivariate version of the Tauberian inversion, the paper shows how to extract precise large deviation principles from double Laplace transforms for processes in the domain of attraction of spectrally positive stable laws. This is illustrated through examples of random sums with correlated increments and stopping times, and for observables of random walks that stay positive.

What carries the argument

The double-transform Tauberian inversion technique, which extends single-variable Tauberian theorems to bivariate Laplace-Stieltjes transforms to obtain asymptotic expansions for large deviations.

Load-bearing premise

The stochastic processes must belong to the domain of attraction of spectrally positive stable laws, otherwise the double-transform inversion and derived asymptotics do not hold.

What would settle it

Derive the large deviation rate function for a process explicitly known not to be in the domain of attraction of spectrally positive stable laws and check if it matches the predictions from the double-transform method.

read the original abstract

In many stochastic models, the observables of interest are naturally encoded in double transforms (e.g., Laplace transforms) that couple spatial and temporal variables. Notably, the double transform often provides the only analytically tractable starting point for the study of processes with correlated increments or path constraints. We extend the Tauberian approach for precise large deviations of stochastic processes belonging to the domain of attraction of spectrally positive stable laws, previously developed for single-variable Laplace--Stieltjes transforms [9], to the bivariate setting. This methodology provides a direct route to asymptotic behaviour that is otherwise difficult to characterize using single-transform techniques. As illustrative examples, we derive precise large deviations for random sums with increments correlated to the stopping time and for path-dependent observables of random walks constrained to remain positive.

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

1 major / 2 minor

Summary. The manuscript extends the Tauberian approach for precise large deviations of stochastic processes in the domain of attraction of spectrally positive stable laws from single-variable Laplace-Stieltjes transforms to the bivariate (double-transform) setting. It derives asymptotic results for random sums whose increments are correlated with the stopping time and for path-dependent observables of random walks constrained to remain positive, using the double transform as the starting point.

Significance. If the bivariate extension holds with the required joint regularity, the method would provide a direct route to precise large-deviation asymptotics in models where double transforms are the natural or only tractable object, particularly those involving correlations or path constraints; this would strengthen the applicability of Tauberian techniques in probability theory beyond the univariate case.

major comments (1)
  1. [Main theorem and random-sum example (§3)] The central extension (presumably stated in the main theorem of §2 or §3) claims that the single-transform inversion carries over to the double-transform setting under the marginal domain-of-attraction condition alone. However, the random-sum example introduces correlation between increments and stopping time, which may induce singularities or prevent uniform convergence in a wedge; the manuscript does not supply an explicit verification or additional hypothesis (e.g., monotonicity in both variables or dominated convergence for the bivariate inversion) that would guarantee the joint tail asymptotics follow. This assumption is load-bearing for the claimed precision of the large-deviation results.
minor comments (2)
  1. [Introduction and §2] Notation for the double Laplace-Stieltjes transform and the precise form of the target asymptotic (e.g., the exact power or logarithmic correction) should be stated uniformly in the introduction and in the statement of the main result to improve readability.
  2. [§1] The reference to the prior single-transform work [9] is appropriate, but a brief recap of the univariate inversion formula would help readers see exactly which step is being generalized.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough review and constructive criticism of our manuscript on the double-transform Tauberian method. The primary concern regarding the justification for the bivariate extension in the presence of correlations is addressed in the point-by-point response below. We have revised the manuscript to incorporate additional details and verifications as suggested.

read point-by-point responses

  1. Referee: [Main theorem and random-sum example (§3)] The central extension (presumably stated in the main theorem of §2 or §3) claims that the single-transform inversion carries over to the double-transform setting under the marginal domain-of-attraction condition alone. However, the random-sum example introduces correlation between increments and stopping time, which may induce singularities or prevent uniform convergence in a wedge; the manuscript does not supply an explicit verification or additional hypothesis (e.g., monotonicity in both variables or dominated convergence for the bivariate inversion) that would guarantee the joint tail asymptotics follow. This assumption is load-bearing for the claimed precision of the large-deviation results.

    Authors: We thank the referee for highlighting this important point. While the main theorem in Section 2 incorporates joint regularity conditions on the double transform (including monotonicity and analyticity in a suitable wedge) that permit the bivariate inversion from the marginal domain-of-attraction assumption, we acknowledge that the correlated random-sum application in Section 3 would benefit from more explicit verification. In the revised manuscript we have added a supporting lemma that confirms the absence of singularities and establishes the required uniform convergence via a dominated-convergence argument adapted to the bivariate Laplace transform, thereby justifying the joint tail asymptotics under the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

Bivariate Tauberian extension introduces independent double-transform inversion without reducing to prior inputs by construction

full rationale

The paper's derivation begins with the standing assumption that the processes belong to the domain of attraction of spectrally positive stable laws and then develops a bivariate Laplace-Stieltjes transform version of the Tauberian inversion previously available only in the single-variable case. This extension is applied to obtain precise large-deviation asymptotics for random sums with correlated increments and for positivity-constrained path functionals. No equation or step in the abstract or described chain equates an output quantity to a fitted parameter or to the input domain condition itself; the joint transform analysis supplies new content for the bivariate setting rather than renaming or recycling the univariate results. The cited single-variable foundation [9] functions as external scaffolding, not as a load-bearing self-reference that collapses the new claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters or invented entities are mentioned. The central domain assumption is recorded below.

axioms (1)
  • domain assumption Stochastic processes belong to the domain of attraction of spectrally positive stable laws.
    Stated in the abstract as the setting in which the double-transform Tauberian method is developed.

pith-pipeline@v0.9.0 · 5657 in / 1269 out tokens · 47087 ms · 2026-05-21T02:19:17.871501+00:00 · methodology

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Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Alili and R

    L. Alili and R. A. Doney. Wiener-Hopf factorization revisited and some applications.Stoch. Stoch. Rep., 66:87–102, 1999

    work page 1999

  2. [2]

    Alili and R

    L. Alili and R. A. Doney. Martin boundaries associated with a killed random walk.Ann. Inst. H. Poincar´ e Probab. Statist., 37(3):313–338, 2001

    work page 2001

  3. [3]

    E. Barkai. Aging in subdiffusion generated by a deterministic dynamical system.Phys. Rev. Lett., 90:104101, 2003

    work page 2003

  4. [4]

    Bianchi, G

    A. Bianchi, G. Cristadoro, and G. Pozzoli. Ladder costs for random walks in L´ evy random media.Stoch. Process. Their Appl., 188:104666, 2025

    work page 2025

  5. [5]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels.Regular Variation. Cambridge University Press, Cambridge, 1987

    work page 1987

  6. [6]

    J. Butt, N. Georgiou, and E. Scalas. Queuing models with Mittag-Leffler inter-event times. Fract. Calc. Appl. Anal., 26:1465–1503, 2023

    work page 2023

  7. [7]

    D. B. H. Cline and T. Hsing. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. 1991. Preprint, Texas A&M University

    work page 1991

  8. [8]

    Cram´ er

    H. Cram´ er. Sur un nouveau th´ eor` eme-limite de la th´ eorie des probabilit´ es.Actualit´ es Sci. Indust., 736:2–23, 1938

    work page 1938

  9. [9]

    Cristadoro and G

    G. Cristadoro and G. Pozzoli. Precise large deviations through a uniform Tauberian theorem. Stoch. Process. Their Appl., 199:104992, 2026

    work page 2026

  10. [10]

    De Haan, E

    L. De Haan, E. Omey, and S. Resnick. Domains of attraction and Regular Variation inR d.J. Multivar. Anal., 14:17–33, 1984

    work page 1984

  11. [11]

    P. Diamond. Slowly varying functions of two variables and a tauberian theorem for the double laplace transform.Appl. Anal., 23(4):301–318, 1987

    work page 1987

  12. [12]

    R. A. Doney. Local behaviour of first passage probabilities.Probab. Theory Relat. Fields, 152:559–588, 2012

    work page 2012

  13. [13]

    R. A. Doney and E. M. Jones. Large deviation results for random walks conditioned to stay positive.Electron. Commun. Probab., 17(38):1–11, 2012

    work page 2012

  14. [14]

    Eliazar and J

    I. Eliazar and J. Klafter. On the first passage of one-sided L´ evy motions.Physica A, 336(3– 4):219–244, 2004

    work page 2004

  15. [15]

    Feller.An Introduction to Probability Theory and its Applications, volume II

    W. Feller.An Introduction to Probability Theory and its Applications, volume II. John Wiley & Sons, Inc., New York, second edition, 1971

    work page 1971

  16. [16]

    Godr` eche and J

    C. Godr` eche and J. M. Luck. Statistics of the occupation time of renewal processes.J. Stat. Phys., 104:489–524, 2001

    work page 2001

  17. [17]

    Gut.Stopped Random Walks

    A. Gut.Stopped Random Walks. Springer-Verlag, New York, second edition, 2009

    work page 2009

  18. [18]

    Gut and S

    A. Gut and S. Janson. The limiting behaviour of certain stopped sums and some applications. Scand. J. Statist., 10:281–292, 1983

    work page 1983

  19. [19]

    G. H. Hardy.Divergent Series. Oxford University Press, Oxford, 1949

    work page 1949

  20. [20]

    I. A. Ibragimov and Y. V. Linnik.Independent and stationary sequences of random variables. Wolters–Noordhoff, Groningen, 1971

    work page 1971

  21. [21]

    T. Kaijser. A stochastic model describing the water motion in a river.Nordic hydrology, II:243–265, 1971. 24 GIAMPAOLO CRISTADORO 1 AND GAIA POZZOLI 1

    work page 1971

  22. [22]

    Koren, V

    T. Koren, V. Chechkin, and J. Klafter. On the first passage time and leapover properties of L´ evy motions.Physica A, 379:10–22, 2007

    work page 2007

  23. [23]

    Koren, A

    T. Koren, A. Lomholt, V. Chechkin, J. Klafter, and R. Metzler. Leapover lenghts and first passage time statistics for L´ evy Flights.Phys. Rev. Lett., 99:160602, 2007

    work page 2007

  24. [24]

    B. B. Mandelbrot.The Fractal Geometry of Nature. W. H. Freeman and Company, New York, 1982

    work page 1982

  25. [25]

    Margolin

    G. Margolin. Nonergodicity of a time series obeying L´ evy statistics.J. Stat. Phys., 122:137– 167, 2006

    work page 2006

  26. [26]

    Omey and E

    E. Omey and E. Willekens. Abelian and Tauberian theorems for the Laplace transform of functions in several variables.J. Multivar. Anal., 30:292–306, 1989

    work page 1989

  27. [27]

    Schumer, D

    R. Schumer, D. A. Benson, and M. M. Meerschaert. Fractal mobile/immobile solute transport. Water Resour. Res., 39(10):1296, 2003

    work page 2003

  28. [28]

    M. F. Shlesinger and J. Klafter. L´ evy walks versus L´ evy flights. In H. E. Stanley and N. Os- trowsky, editors,On Growth and Form: Fractal and Non-Fractal Patterns in Physics, volume 1837 ofNATO ASI Series E: Applied Sciences, pages 279–283. Martinus Nijhoff Publihers, Lancaster, 1986

    work page 1986

  29. [29]

    D. V. Widder.The Laplace Transform. Princeton University Press, Princeton, 1946. 1Dipartimento di Matematica e Applicazioni, Universit `a degli Studi di Milano- Bicocca, Via R. Cozzi 55, 20125 Milano, Italy. Email address:giampaolo.cristadoro@unimib.it Email address:gaia.pozzoli@unimib.it

    work page 1946