Pith. sign in

REVIEW 2 major objections 3 minor 1 cited by

On the multidimensional elephant random walk with stops

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Pausing elephant walk obeys a Mittag-Leffler limit in any dimension.

desk verdict The Gram matrix limit with the Mittag-Leffler distribution is new and likely correct, but the LIL theorems are off by a factor d from an invalid coordinate-wise sum; the a.s. and CLT results look sound, and the paper is worth refereeing after corrections. read the letter →

arxiv 2501.14594 v1 pith:Z4N3WBPM submitted 2025-01-24 math.PR math.STstat.TH

Bernard Bercu This is my paper · ORCID
classification math.PRmath.STstat.TH MSC 60G5060G4260F05
keywords elephantrandomwalkwithstopsmultidimensionalMittag-Lefflerdistributionmartingalestronglawoflargenumberstheiteratedlogarithmasymptoticnormality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the one-dimensional elephant random walk with stops to every dimension $d \ge 1$, allowing the walker to stay put with probability $r$. Its central claim is that the Gram matrix of the walk's step vectors, normalized by $n^{1-r}$, converges almost surely to $\frac{1}{d}\Sigma I_d$, where $\Sigma$ is a Mittag-Leffler random variable with parameter $1-r$. From that single limit, the paper derives a complete asymptotic picture: strong laws in the diffusive and critical regimes, a law of the iterated logarithm, self-normalized central limit theorems, and a superdiffusive limit with Gaussian fluctuations. A curious reader would care because the elephant random walk is a simple non-Markovian model of long-range memory, and allowing stops is the natural way to let the walker hesitate without changing the model's spirit.

What carries the argument

The central object is the Gram matrix $\Sigma_n = \sum_{k=1}^n X_k X_k^{\mathsf T}$ and its trace $\sigma_n^2 = \operatorname{Tr}(\Sigma_n)$; the walk's increments are the vectors $X_k$. Two martingales carry the proof: $N_n = b_n \sigma_n^2$, whose conditional dynamics $E[\sigma_{n+1}^2 \mid \mathcal{F}_n] = (1+b/n)\sigma_n^2$ with $b=1-r$ produce the Mittag-Leffler limit, and $M_n = a_n S_n$ with $a_n = \Gamma(n)\Gamma(a+1)/\Gamma(n+a)$, $a=p-q$, whose predictable quadratic variation splits as $\langle M\rangle_n = \frac{1}{d}I_d + V_n - a^2 W_n$. The Mittag-Leffler distribution $\mathrm{ML}(\alpha)$ is the positive random variable with Laplace transform $E_\alpha(t)=\sum_{n\ge0} t^n/\Gamma(1+n\alpha)$; it is identified here by its moments $E[X^m]=m!/\Gamma(1+m\alpha)$, a characterization that the paper invokes to ensure the distribution is determined by its moments. The fourth-moment bound on the martingale increments is what lets the martingale limit theorems be applied.

What would settle it

For $d=2$ and a diffusive choice of $p$, compute $E[\|\varepsilon_{n+1}\|^4 \mid \mathcal{F}_n]$ directly from the displayed formulas (A.11)-(A.12) for a sequence of $n$, and test the inequality against $7b\sigma_n^2/n$; a single violation would falsify the bound, and a Monte Carlo estimate of the ratio would provide numerical evidence either way. An algebraic derivation from (A.12) yielding a larger prefactor would also disprove the stated bound.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is Lemma 2.1: for any $p,q \in [0,1]$ and $r \in (0,1)$, $\lim_{n\to\infty} n^{-(1-r)} \Sigma_n = \frac{1}{d}\Sigma I_d$ almost surely and in every $L^m$, where $\Sigma$ has the Mittag-Leffler distribution with parameter $1-r$. The trace version gives $\sigma_n^2 / n^{1-r} \to \Sigma$ almost surely. The paper then shows that in the diffusive regime $p < p_{d,r} = \frac{2d+1}{4d}(1-r)$, the position satisfies $S_n/n \to 0$, the law of the iterated logarithm $\limsup \|S_n\|^2/(2\sigma_n^2 \log\log\sigma_n^2) = d v^2$, and $S_n/\sqrt{\sigma_n^2} \to N(0, v^2 I_d)$; in the critical regime $p = p_{d,r}$, the same results hold with extra $\log$ factors and covariance $\frac{1}{d}I_d$; in the superdiffusive regime $p > p_{d,r}$, $S_n/n^{p-q} \to L$ almost surely with explicit covariance, and $(S_n - n^{p-q}L)/\sqrt{\sigma_n^2} \to N(0, \vartheta^2 I_d)$.

Load-bearing premise

The load-bearing premise is the conditional fourth-moment bound $E[\|\varepsilon_{n+1}\|^4 \mid \mathcal{F}_n] \le 7b\sigma_n^2/n$ in Eq. (A.13), which the paper states as the outcome of tedious but straightforward calculations without displaying the full algebra; if that bound fails, the Lindeberg-type conditions and the convergence of the auxiliary martingales used in the limit theorems would not follow.

Editorial extensions

If this is right

  • The critical value $p_{d,r}=\frac{2d+1}{4d}(1-r)$ fully separates the regimes; below it the walk is diffusive, at it the normalization picks up logarithmic factors, and above it a nondegenerate random limit $L$ exists.
  • Self-normalization by $\sigma_n^2$ removes the Mittag-Leffler randomness, giving pure Gaussian limits in the diffusive and critical regimes; normalization by $n^{1-r}$ instead yields Gaussian mixtures with Mittag-Leffler variance.
  • The universal almost sure limit $\sigma_n^2/n^{1-r}\to\Sigma$ holds for all parameters and all dimensions, making the squared displacement's random scaling law explicit.
  • In the superdiffusive regime, the fluctuation of $S_n$ around its random limit $L$, normalized by $\sqrt{\sigma_n^2}$, is asymptotically Gaussian with covariance $\vartheta^2 I_d$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension not in the paper: because the limiting Gram matrix is isotropic, the same limit theorems would likely hold for any initial step distribution with covariance proportional to $I_d$, not only for the uniform choice over the $2d$ directions.
  • Extension not in the paper: other additive functionals of the walk, such as the center of mass, should show the same Mittag-Leffler mixing under the $n^{1-r}$ normalization; checking this is a natural test of the mechanism.
  • Extension not in the paper: the fourth-moment bound (A.13) is asserted after 'tedious but straightforward calculations'; a reader seeking weaker hypotheses would need a different path than the martingale limit theorems used here.
  • Extension not in the paper: if the stay probability or the direction probabilities were made coordinate-dependent, the deterministic factor $\frac{1}{d}I_d$ would become some anisotropic positive semidefinite matrix; the present theorem shows only the isotropic case.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies the multidimensional elephant random walk with stops (MERWS), a d-dimensional process in which, at each step, a uniformly chosen past increment is repeated with probability p, changed to one of the other 2d-1 directions with probability q, or set to zero with probability r. The main result (Lemma 2.1) states that the Gram matrix of the increments, normalized by n^{1-r}, converges almost surely to (1/d) Σ I_d, where Σ has an ML(1-r) distribution. Building on this, the author proves almost sure convergence of the position in the diffusive and critical regimes, a law of the iterated logarithm, self-normalized central limit theorems, and in the superdiffusive regime almost sure convergence to a nondegenerate limit with Gaussian fluctuations around it. The proofs use multidimensional martingale techniques and generalize the author's one-dimensional results.

Significance. The paper is well motivated and its martingale approach is natural; Lemma 2.1 is a genuine extension of the one-dimensional trace result, and the almost sure convergence and CLT statements are plausible and largely checkable. The derivations are detailed, and the paper relies on a published, parameter-free lemma from [3] for the one-dimensional martingale convergence, which is independent of the multidimensional target and not circular. However, the laws of the iterated logarithm as stated in Theorems 3.2 and 3.5 are incorrect: the norm constants are off by a factor d. This is a substantial error in the main results, and the paper needs revision.

major comments (2)
  1. [Theorem 3.2 and Appendix A.2 (Eqs. (A.20)-(A.21))] The norm law of the iterated logarithm is stated with the wrong constant. The proof passes from the directional LIL (A.20), which gives limsup_n ⟨u,S_n⟩^2/(2σ_n^2 log log σ_n^2) = v^2||u||^2 for each fixed u, to the norm statement (A.21) by summing the d coordinate inequalities. This step is invalid because the limsup of a sum of nonnegative sequences is not the sum of the limsups; for an isotropic vector with asymptotic covariance v^2 I_d, the correct norm LIL constant is v^2, not d v^2. The standard epsilon-net/compactness argument applied to (A.20) yields limsup_n ||S_n||^2/(2σ_n^2 log log σ_n^2) = v^2 a.s. Consequently, (3.4) should read v^2, and (3.5) should read v^2 Σ, not d v^2 Σ. The equality in (A.22) identifies b/(b-2a) with d v^2, which is the sum of the d per-coordinate constants rather than the norm constant.
  2. [Theorem 3.5 and Appendix B.2] The same factor-d error appears in the critical regime. The directional LIL in the proof gives, for every unit vector u, limsup_n ⟨u,S_n⟩^2/(2σ_n^2 log n log log n) = 2a/d a.s. (since b=2a). The compactness argument then gives the same constant for the norm: limsup_n ||S_n||^2/(2σ_n^2 log n log log n) = 2a/d a.s. Because log σ_n^2 = 2a log n (1+o(1)), this implies limsup_n ||S_n||^2/(2σ_n^2 log σ_n^2 log log log σ_n^2) = 1/d, not 1 as stated in (3.10). Similarly, the constant in (3.11) should be ((1-r)/d)Σ, not (1-r)Σ. The proof omits the conversion step and directly asserts (3.10), which is where the factor d enters.
minor comments (3)
  1. [Appendix A.2, Eq. (A.13)] The fourth-moment bound (A.13) is asserted after a 'tedious but straightforward calculation' without the intermediate algebra. Since this bound is used to verify Heyde's conditions for the LIL and CLT, the authors should provide a more detailed derivation, at least in an appendix.
  2. [Various] There are several typographical issues: the title in the header reads 'W ALK'; a reference to a monograph on Mittag-Leffler functions appears as '[ ?]' without an entry; and equation (A.12) contains a suspicious '+ -4' sign. These should be corrected.
  3. [Appendix B.3, Eq. (3.13)] In the proof of Theorem 3.6, the derivation of (3.13) is left to the reader; providing the details would increase the paper's verifyability, especially given the delicate conversions used elsewhere.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the multidimensional Gram-matrix limit and the regime theorems are derived from the model recursion and an independent one-dimensional trace result; no prediction reduces to a fitted input or to the target result.

full rationale

The paper's derivation chain is self-contained apart from Lemma 4.1, which imports the one-dimensional trace martingale convergence from the author's prior paper [3]. That import is not circular: it is a published, parameter-free statement about the scalar martingale N_n=b_nσ_n^2 with b=1−r, and its assumptions do not include the multidimensional Gram-matrix limit (2.4). The paper verifies that the trace σ_n^2 satisfies the same recursion (4.12) in any dimension, so Lemma 4.1 applies directly; the genuinely new content, namely the coordinate-wise limits σ_n^2(i)/n^b → Σ/d and hence Σ_n/n^b → (1/d)Σ I_d, is then derived via the martingale decompositions (4.22)–(4.23), Toeplitz's lemma, and the strong law of large numbers for martingales. The regime theorems (Theorems 3.1–3.8) follow from this Gram-matrix limit and standard martingale LIL/CLT criteria; no parameter is fitted to data and no target result appears as an assumption. One could note the self-citation to [3] as a provenance remark, but under the stated rules it counts as independent support rather than circularity. The skeptical concern about the norm LIL constant dv^2 is a mathematical correctness issue about summing coordinate-wise limsups, not a circularity of the derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model parameters p, q, r are inputs, not fitted. The proof relies on standard martingale limit theorems and on one imported lemma from the author's previous paper [3] whose content (the trace Mittag-Leffler convergence) is a published independent result. No new entities are postulated.

assumptions (4)
  • domain assumption Lemma 4.1 from [3]: the martingale N_n = b_n σ²_n is bounded in L^m and converges to N with moments m!(Γ(b+1))^m/Γ(1+mb), yielding the trace Mittag-Leffler convergence.
    Imported without proof from the author's earlier one-dimensional elephant random walk with stops paper. It is a published parameter-free theorem, not derived in this paper. It is used in the proof of Lemma 2.1 to identify the normalization n^{1-r} and the Mittag-Leffler distribution.
  • standard math Heyde's martingale LIL and CLT theorems [17] (Theorem 1, Corollaries 1 and 2).
    Used to prove Theorems 3.2, 3.3, 3.5, 3.6, and 3.8 after verifying predictable variation and Lindeberg-type conditions.
  • standard math Standard strong law of large numbers for martingales (Theorems 1.3.24 and 4.3.15/4.3.16 in Duflo [12]) and Doob's martingale convergence theorem [16].
    Used in Appendices A, B, C to bound M_n and N_n(i).
  • standard math Toeplitz's lemma and Wendel's inequality for gamma ratios.
    Used repeatedly to convert Cesàro averages of asymptotically equivalent sequences.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the multidimensional elephant random walk with stops." pith.science (2026). https://pith.science/paper/Z4N3WBPM

@misc{pith2026250114594,
  author       = {Pith},
  title        = {Pith review of: On the multidimensional elephant random walk with stops},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4N3WBPM}},
  note         = {Machine review of arXiv:2501.14594}
}
read the original abstract

The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We prove that the Gram matrix associated with the MERWS, properly normalized, converges almost surely to the product of a deterministic matrix, related to the axes on which the MERWS moves uniformly, and a Mittag-Leffler distribution. It allows us to extend all the results previously established for the one-dimensional elephant random walk with stops. More precisely, in the diffusive and critical regimes, we prove the almost sure convergence of the MERWS. In the superdiffusive regime, we establish the almost sure convergence of the MERWS, properly normalized, to a nondegenerate random vector. We also study the self-normalized asymptotic normality of the MERWS.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A model of opinion dynamics evolving via a preferential attachment mechanism involving multiple extractions

    math.PR 2026-08 conditional novelty 6.0 of 10

    For a two-opinion preferential-attachment network with multiple sampling and general reinforcement, the normalized opinion count, influence capital and activity converge almost surely to invariant sets of a mean-field...

Reference graph

Works this paper leans on

22 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [3]

    B. Bercu. On the elephant random walk with stops playing hide and s eek with the Mittag- Leffler distribution. J. Stat. Phys. , 189(1):Paper No. 12, 27, 2022

    work page 2022

  2. [1]

    Baur and J

    E. Baur and J. Bertoin. Elephant random walks and their connect ion to p´ olya-type urns. Physical review. E 94, 052134 , 2016

    work page 2016

  3. [2]

    B. Bercu. A martingale approach for the elephant random walk. J. Phys. A , 51(1):015201, 16, 2018

    2018

  4. [4]

    Bercu and L

    B. Bercu and L. Laulin. On the multi-dimensional elephant random w alk. J. Stat. Phys. , 175(6):1146–1163, 2019

    work page 2019

  5. [5]

    Bercu and L

    B. Bercu and L. Laulin. On the center of mass of the elephant ran dom walk. Stochastic Process. Appl., 133:111–128, 2021

    work page 2021

  6. [6]

    Bertenghi

    M. Bertenghi. Functional limit theorems for the multi-dimensional elephant random walk. Stoch. Models, 38(1):37–50, 2022

    work page 2022

  7. [7]

    J. Bertoin. Scaling exponents of step-reinforced random walks . Probability Theory and Related Fields, 179(1):295–315, 2021

    work page 2021

  8. [8]

    J. Bertoin. Counting the zeros of an elephant random walk. Trans. Amer. Math. Soc. , 375(8):5539–5560, 2022

    work page 2022

Show all 22 references
  1. [9]

    Chen and L

    J. Chen and L. Laulin. Analysis of the smoothly amnesia-reinforce d multidimensional elephant random walk. J. Stat. Phys. , 190(10):Paper No. 158, 42, 2023

    2023

  2. [10]

    C. F. Coletti, R. Gava, and G. M. Sch¨ utz. Central limit theorem and related results for the elephant random walk. J. Math. Phys. , 58(5):053303, 8, 2017

    2017

  3. [11]

    Dedecker, X

    J. Dedecker, X. Fan, H. Hu, and F. Merlev` ede. Rates of conv ergence in the central limit theorem for the elephant random walk with random step sizes. J. Stat. Phys. , 190(10):Paper No. 154, 30, 2023

    2023

  4. [12]

    M. Duflo. Random iterative models , volume 34 of Applications of Mathematics (New York) . Springer-Verlag, Berlin, 1997

    1997

  5. [13]

    X. Fan, H. Hu, and X. Ma. Cram´ er moderate deviations for the elephant random walk. J. Stat. Mech. Theory Exp. , (2):Paper No. 023402, 20, 2021

    2021

  6. [14]

    Gonz´ alez-Navarrete

    M. Gonz´ alez-Navarrete. Multidimensional walks with random te ndency. J. Stat. Phys. , 181(4):1138–1148, 2020

    2020

  7. [15]

    Gut and U

    A. Gut and U. Stadtm¨ uller. Variations of the elephant random w alk. J. Appl. Probab. , 58(3):805–829, 2021

    2021

  8. [16]

    Hall and C

    P. Hall and C. C. Heyde. Martingale limit theory and its application . Academic Press, Inc., New York-London, 1980. Probability and Mathematical Statistics

    1980

  9. [17]

    C. C. Heyde. On central limit and iterated logarithm supplements to the martingale conver- gence theorem. Journal of Applied Probability , 14(4):758–775, 1977

    1977

  10. [18]

    Z. Hu, W. Wang, and L. Dong. Strong approximations in the almos t sure central limit theorem and limit behavior of the center of mass. Stochastic Process. Appl. , 182:104570, 2025

    2025

  11. [19]

    Kiss and B

    J. Kiss and B. Veto. Moments of the superdiffusive elephant ran dom walk with general step distribution. Electron. Commun. Probab., 27:Paper No. 44, 12, 2022

    2022

  12. [20]

    Kubota and M

    N. Kubota and M. Takei. Gaussian fluctuation for superdiffusive elephant random walks. J. Stat. Phys. , 177(6):1157–1171, 2019

    2019

  13. [21]

    R. Roy, M. Takei, and H. Tanemura. The elephant random walk in t he triangular array setting. Journal of Applied Probability , To appear, 2025

    2025

  14. [22]

    G. M. Sch¨ utz and S. Trimper. Elephants can always remember: Exact long-range memory effects in a non-markovian random walk. Physical review. E 70, 045101 , 2004. Email address : bernard.bercu@math.u-bordeaux.fr Universit´ e de Bordeaux, Institut de Math ´ ematiques de Bordeau...

    2004

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.