pith. sign in

arxiv: 2605.17725 · v1 · pith:MBHPHYDZnew · submitted 2026-05-18 · 🧮 math.PR · cond-mat.stat-mech· math-ph· math.MP

Limit theorems for random walks with spatio-temporal drift

Ngo P.N. Ngoc , Tuan-Minh Nguyen This is my paper

Pith reviewed 2026-05-19 21:55 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.stat-mechmath-phmath.MP
keywords random walkslimit theoremsspatio-temporal driftself-interacting processesphase transitionGaussian convergencestochastic differential equationslocalization
0
0 comments X

The pith

Random walks with position-and-time dependent polynomial drift show three distinct asymptotic regimes.

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

The paper examines discrete-time random walks in d-dimensional space whose drift at each step weakens polynomially with elapsed time yet strengthens polynomially with distance from the origin. Under the standing assumption that increments possess moments of order greater than two, the authors derive the long-term scaling behavior. In the linear-drift setting they locate a phase transition that governs whether the suitably normalized position converges in distribution to a Gaussian. In the nonlinear setting they partition the parameter space along a critical line into three regimes whose normalized processes converge either to a Gaussian, to the stationary law of a limiting stochastic differential equation, or localize almost surely on a hypersphere.

Core claim

In the nonlinear case, the authors identify three distinct regimes separated by a critical line and show that the normalized process exhibits qualitatively different behaviors in each regime, including convergence in distribution to a Gaussian law, convergence to a non-Gaussian limit given by the stationary distribution of a stochastic differential equation, and almost sure localization on a hypersphere. In the linear case a phase transition governs convergence to Gaussian limits.

What carries the argument

The conditional drift that decays polynomially in time and grows polynomially with distance from the origin, which governs the scaling and the form of the limiting object.

If this is right

  • In the linear case the convergence in distribution of the normalized process to a Gaussian undergoes a phase transition at a critical parameter value.
  • In one nonlinear regime the normalized process converges in distribution to a Gaussian law.
  • In a second nonlinear regime the normalized process converges in distribution to the stationary distribution of an associated stochastic differential equation.
  • In the third nonlinear regime the walk localizes almost surely on a hypersphere.

Where Pith is reading between the lines

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

  • The same regime diagram may appear in continuous-time self-interacting diffusions whose drift is allowed to weaken with time.
  • Numerical sampling across the critical line could locate the precise location of the transition and test the predicted change in fluctuation scale.
  • Localization on a hypersphere implies that the long-run radial coordinate remains bounded while angular motion continues.
  • Higher-moment assumptions or moment-free conditions could be examined to see whether the same trichotomy persists.

Load-bearing premise

The increments of the random walk possess finite moments of order strictly greater than two.

What would settle it

Numerical simulation of the walk for concrete polynomial exponents on either side of the critical line, checking whether the empirical distribution of the scaled position matches a Gaussian, matches the stationary measure of the predicted SDE, or concentrates on a sphere.

read the original abstract

We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models of self-interacting random processes. We determine the asymptotic behavior of the walk under the assumption that its increments have moments of order $p$ for some $p>2$. In the linear case, where the drift depends linearly on the current position, we establish a phase transition in the convergence in distribution of the normalized process to Gaussian limits. In the nonlinear case, we identify three distinct regimes separated by a critical line and show that the normalized process exhibits qualitatively different behaviors in each regime, including convergence in distribution to a Gaussian law, convergence to a non-Gaussian limit given by the stationary distribution of a stochastic differential equation, and almost sure localization on a hypersphere.

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 studies discrete-time random walks in R^d whose conditional drift decays polynomially in time and grows polynomially with distance to the origin. Under the standing assumption that increments have moments of order p>2, the authors establish asymptotic results: a phase transition for normalized convergence in distribution to Gaussian limits in the linear case, and in the nonlinear case a trichotomy of regimes separated by a critical line, with behaviors including distributional convergence to a Gaussian, convergence to the stationary distribution of an SDE, and almost sure localization on a hypersphere.

Significance. If the derivations hold, the work provides a detailed phase diagram for the long-term behavior of self-interacting random walks with spatio-temporal polynomial drifts, extending known models in this area. The distinction between linear and nonlinear settings and the explicit identification of three qualitatively different regimes (including the a.s. localization) would constitute a solid contribution to the literature on random processes with memory or position-dependent interactions, particularly when the moment condition is shown to be sufficient across all regimes.

major comments (1)
  1. [Abstract / nonlinear regimes] Abstract (nonlinear case): The almost sure localization on a hypersphere is claimed under the moment assumption p>2. This is load-bearing for the trichotomy, yet with only polynomial moments of order p>2 the position-dependent nonlinear drift may amplify the effect of large jumps, potentially yielding positive probability of escape from any fixed-radius shell. The manuscript should supply the concrete argument (e.g., the Borel-Cantelli lemma or tail-control estimate) that establishes why p>2 is nevertheless adequate for the a.s. statement.
minor comments (2)
  1. [Main theorems] The normalization factor applied to the process should be stated explicitly when the main theorems are formulated, rather than left implicit in the abstract.
  2. [Nonlinear case] Clarify the precise definition of the critical line that separates the three regimes in the nonlinear case, including how it depends on the polynomial exponents.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point below and will incorporate revisions to strengthen the presentation of the almost sure localization result.

read point-by-point responses

  1. Referee: [Abstract / nonlinear regimes] Abstract (nonlinear case): The almost sure localization on a hypersphere is claimed under the moment assumption p>2. This is load-bearing for the trichotomy, yet with only polynomial moments of order p>2 the position-dependent nonlinear drift may amplify the effect of large jumps, potentially yielding positive probability of escape from any fixed-radius shell. The manuscript should supply the concrete argument (e.g., the Borel-Cantelli lemma or tail-control estimate) that establishes why p>2 is nevertheless adequate for the a.s. statement.

    Authors: We agree that the justification for the almost sure localization under the standing moment assumption p>2 merits a more explicit and self-contained presentation to address potential concerns about large jumps being amplified by the nonlinear drift. In the current proof of the localization statement, moment bounds are used to obtain summable tail probabilities for jumps exceeding the current radius, which are then fed into a Borel-Cantelli argument; the time-decaying component of the drift further ensures that the relevant events become rare enough for the sum of probabilities to converge. Nevertheless, we acknowledge that this chain of estimates is not spelled out with sufficient clarity in the manuscript. In the revised version we will add a dedicated remark (or short lemma) that isolates the tail-control estimate and its application to the a.s. property, making transparent why p>2 is sufficient to preclude escape from the hypersphere with probability one. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no circular reductions identified

full rationale

The paper derives limit theorems for random walks with spatio-temporal drift from the standing assumption that increments possess moments of order p>2, applying standard tools from stochastic processes to obtain phase transitions, distributional convergences, and almost-sure localization statements in the nonlinear regime. No self-definitional steps appear where a claimed output is defined in terms of itself, no fitted parameters are relabeled as independent predictions, and no load-bearing self-citations or imported uniqueness theorems reduce the central claims to prior author work by construction. The trichotomy of regimes follows directly from the moment condition and the form of the drift without circular equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of moments of order p>2 for the increments and on standard background results in probability theory for tightness and convergence of processes; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Increments possess moments of order p for some p>2
    Explicitly stated in the abstract as the condition under which all asymptotic results hold.

pith-pipeline@v0.9.0 · 5691 in / 1376 out tokens · 52918 ms · 2026-05-19T21:55:19.303749+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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

37 extracted references · 37 canonical work pages

  1. [1]

    Probability6(1978), no

    David Aldous,Stopping times and tightness, Ann. Probability6(1978), no. 2, 335–340. MR 474446

    work page 1978

  2. [2]

    1709, Springer, Berlin, 1999, pp

    Michel Bena ¨ım,Dynamics of stochastic approximation algorithms, S ´eminaire de Probabilit ´es, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 1–68. MR 1767993

    work page 1999

  3. [3]

    Bernard Bercu,A martingale approach for the elephant random walk, J. Phys. A51(2018), no. 1, 015201, 16. MR 3741953

    work page 2018

  4. [4]

    Bernard Bercu and Lucile Laulin,On the multi-dimensional elephant random walk, J. Stat. Phys.175(2019), no. 6, 1146–1163. MR 3962977

    work page 2019

  5. [5]

    Nils Berglund,Long-time dynamics of stochastic differential equations, arXiv preprint arXiv:2106.12998 (2021)

    work page arXiv 2021

  6. [6]

    Jean Bertoin,Noise reinforcement for L ´evy processes, Ann. Inst. Henri Poincar ´e Probab. Stat.56(2020), no. 3, 2236–2252. MR 4116724

    work page 2020

  7. [7]

    Theory Related Fields179(2021), no

    ,Scaling exponents of step-reinforced random walks, Probab. Theory Related Fields179(2021), no. 1-2, 295–315. MR 4221659

    work page 2021

  8. [8]

    Celebrating Vladas Sidoravicius, Progr

    ,Universality of noise reinforced Brownian motions, In and out of equilibrium 3. Celebrating Vladas Sidoravicius, Progr. Probab., vol. 77, Birkh¨auser/Springer, Cham, 2021, pp. 147–161. MR 4237267

    work page 2021

  9. [9]

    ,Counting the zeros of an elephant random walk, Trans. Amer. Math. Soc.375(2022), no. 8, 5539–5560. MR 4469228 LIMIT THEOREMS FOR RANDOM W ALKS WITH SPATIO-TEMPORAL DRIFT 45

    work page 2022

  10. [10]

    ,Counterbalancing steps at random in a random walk, J. Eur. Math. Soc. (JEMS)26(2024), no. 7, 2655–2677. MR 4756572

    work page 2024

  11. [11]

    MR 1700749

    Patrick Billingsley,Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication. MR 1700749

    work page 1999

  12. [12]

    Coletti, Renato Gava, and Gunter M

    Cristian F. Coletti, Renato Gava, and Gunter M. Sch ¨utz,Central limit theorem and related results for the elephant random walk, J. Math. Phys.58(2017), no. 5, 053303, 8. MR 3652225

    work page 2017

  13. [13]

    ,A strong invariance principle for the elephant random walk, J. Stat. Mech. Theory Exp. (2017), no. 12, 123207, 8. MR 3748931

    work page 2017

  14. [14]

    Andrea Collevecchio, Tuan-Minh Nguyen, and Stanislav V olkov,Vertex-reinforced jump process on the integers with nonlinear reinforcement, Ann. Appl. Probab.32(2022), no. 4, 2671–2705. MR 4474517

    work page 2022

  15. [15]

    Denis Denisov, Dmitry Korshunov, and Vitali Wachtel,Markov chains with asymptotically zero drift: Lamperti’s problem, New Mathematical Monographs, Cambridge University Press, 2025

    work page 2025

  16. [16]

    23, Springer- Verlag, Berlin, 1996

    Marie Duflo,Algorithmes stochastiques, Math ´ematiques & Applications (Berlin) [Mathematics & Applications], vol. 23, Springer- Verlag, Berlin, 1996. MR 1612815

    work page 1996

  17. [17]

    Ethier and Thomas G

    Stewart N. Ethier and Thomas G. Kurtz,Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986, Characterization and convergence. MR 838085

    work page 1986

  18. [18]

    Mihai Gradinaru and Yoann Offret,Existence and asymptotic behaviour of some time-inhomogeneous diffusions, Ann. Inst. Henri Poincar´e Probab. Stat.49(2013), no. 1, 182–207. MR 3060153

    work page 2013

  19. [19]

    H ´el`ene Gu´erin, Lucile Laulin, Kilian Raschel, and Thomas Simon,On the limit law of the superdiffusive elephant random walk, Electron. J. Probab.30(2025), Paper No. 102, 25. MR 4926011

    work page 2025

  20. [20]

    Mattingly,Yet another look at Harris’ ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI, Progr

    Martin Hairer and Jonathan C. Mattingly,Yet another look at Harris’ ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI, Progr. Probab., vol. 63, Birkh¨auser/Springer Basel AG, Basel, 2011, pp. 109–117. MR 2857021

    work page 2011

  21. [21]

    Hall and C

    P. Hall and C. C. Heyde,Martingale limit theory and its application, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 624435

    work page 1980

  22. [22]

    J ´odar and J

    L. J ´odar and J. C. Cort ´es,Some properties of gamma and beta matrix functions, Appl. Math. Lett.11(1998), no. 1, 89–93. MR 1490386

    work page 1998

  23. [23]

    John Lamperti,Criteria for the recurrence or transience of stochastic process. I, J. Math. Anal. Appl.1(1960), 314–330. MR 126872

    work page 1960

  24. [24]

    ,A new class of probability limit theorems, J. Math. Mech.11(1962), 749–772. MR 148120

    work page 1962

  25. [25]

    ,Criteria for stochastic processes. II. Passage-time moments, J. Math. Anal. Appl.7(1963), 127–145. MR 159361

    work page 1963

  26. [26]

    Menozzi, A

    S. Menozzi, A. Pesce, and X. Zhang,Density and gradient estimates for non degenerate Brownian SDEs with unbounded measur- able drift, Journal of Differential Equations272(2021), 330–369

    work page 2021

  27. [27]

    209, Cambridge University Press, Cambridge, 2017, Lyapunov function methods for near-critical stochastic systems

    Mikhail Menshikov, Serguei Popov, and Andrew Wade,Non-homogeneous random walks, Cambridge Tracts in Mathematics, vol. 209, Cambridge University Press, Cambridge, 2017, Lyapunov function methods for near-critical stochastic systems. MR 3587911

    work page 2017

  28. [28]

    Mikhail Menshikov and Stanislav V olkov,Urn-related random walk with driftρx α/tβ, Electron. J. Probab.13(2008), no. 31, 944–960. MR 2413290

    work page 2008

  29. [29]

    Meyn and R

    Sean P. Meyn and R. L. Tweedie,Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab.25(1993), no. 3, 518–548. MR 1234295

    work page 1993

  30. [30]

    Theory Related Fields158(2014), no

    Yoann Offret,Invariant distributions and scaling limits for some diffusions in time-varying random environments, Probab. Theory Related Fields158(2014), no. 1-2, 1–38. MR 3152778

    work page 2014

  31. [31]

    Mariane Pelletier,Weak convergence rates for stochastic approximation with application to multiple targets and simulated an- nealing, Ann. Appl. Probab.8(1998), no. 1, 10–44. MR 1620405

    work page 1998

  32. [32]

    MR 2941202

    Robin Pemantle,Random processes with reinforcement, ProQuest LLC, Ann Arbor, MI, 1988, Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2941202

    work page 1988

  33. [33]

    Probab.53(2025), no

    Shuo Qin,Recurrence and transience of multidimensional elephant random walks, Ann. Probab.53(2025), no. 3, 1049–1078. MR 4908097

    work page 2025

  34. [34]

    Olivier Raimond and Tuan-Minh Nguyen,Strongly vertex-reinforced jump process on a complete graph, Ann. Inst. Henri Poincar´e Probab. Stat.57(2021), no. 3, 1549–1568. MR 4291445 46 LIMIT THEOREMS FOR RANDOM W ALKS WITH SPATIO-TEMPORAL DRIFT

    work page 2021

  35. [35]

    Olivier Raimond and Pierre Tarres,Non-convergence to unstable equilibriums for continuous-time and discrete-time stochastic processes, arXiv preprint arXiv:2311.02978 (2023)

    work page arXiv 2023

  36. [36]

    Robbins and D

    H. Robbins and D. Siegmund,A convergence theorem for non negative almost supermartingales and some applications, Optimiz- ing methods in statistics (Proc. Sympos., Ohio State Univ., Columbus, Ohio, 1971), Academic Press, New York-London, 1971, pp. 233–257. MR 343355

    work page 1971

  37. [37]

    Sch ¨utz and Steffen Trimper,Elephants can always remember: Exact long-range memory effects in a non-markovian random walk, Phys

    Gunter M. Sch ¨utz and Steffen Trimper,Elephants can always remember: Exact long-range memory effects in a non-markovian random walk, Phys. Rev. E70(2004), 045101

    work page 2004