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arxiv: 2606.21111 · v1 · pith:7UV7TDFZnew · submitted 2026-06-19 · 🧮 math.PR

On Elephant Random Walk with Delayed Amnesia

Manisha Dhillon , Shyan Ghosh , Kuldeep Kumar Kataria This is my paper

Pith reviewed 2026-06-26 13:45 UTC · model grok-4.3

classification 🧮 math.PR
keywords elephant random walkdelayed amnesiavector martingalealmost sure convergencelaw of the iterated logarithmasymptotic normalitymean square displacementcenter of mass
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The pith

A modified elephant random walk transitions from uniform memory to selective amnesia while preserving martingale structure, yielding almost sure convergence and asymptotic normality in diffusive and critical regimes.

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

The paper introduces an elephant random walk variant whose memory rule switches after a delay from recalling all past steps uniformly to recalling them selectively. A vector martingale is built from the position increments under this rule. In the diffusive and critical parameter regimes the martingale yields almost sure convergence of the walk, a law of the iterated logarithm, asymptotic normality after suitable scaling, and the exact growth rate of mean-square displacement; the same conclusions hold for the associated center of mass and for an extension with random step lengths. In the superdiffusive regime only almost sure convergence and the displacement rate are obtained. These limit theorems describe the long-term trajectory of a process whose memory fades in a controlled way.

Core claim

In the diffusive and critical regimes the position of the delayed-amnesia elephant walk converges almost surely, obeys a law of the iterated logarithm, is asymptotically normal, and has mean-square displacement of explicit order; in the superdiffusive regime it converges almost surely with a different displacement rate. The same statements hold for the center of mass of the walk and for the version with random step sizes. All proofs rely on the fact that the chosen memory transition keeps the position process a vector martingale.

What carries the argument

The vector martingale formed by the suitably weighted position process, whose increments remain orthogonal under the delayed transition from uniform to selective memory.

If this is right

  • The walk position converges almost surely in every regime.
  • A law of the iterated logarithm and asymptotic normality hold in the diffusive and critical regimes.
  • Explicit asymptotic rates for mean-square displacement are available in all regimes.
  • Identical limit theorems apply to the center of mass and to the random-step-size extension.

Where Pith is reading between the lines

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

  • The martingale construction may extend directly to other memory-transition functions that preserve orthogonality of increments.
  • The center-of-mass results could be used to compare persistence properties with other persistent random-walk models in one dimension.
  • Numerical verification of the mean-square-displacement exponents would provide an immediate check on the regime boundaries.

Load-bearing premise

The specific functional form of the memory transition is chosen so that the position process remains a vector martingale throughout the diffusive, critical, and superdiffusive regimes.

What would settle it

A simulation in the critical regime in which the normalized position fails to converge in distribution to a non-degenerate Gaussian limit would falsify the asymptotic normality claim.

read the original abstract

In this paper, we introduce a modified elephant random walk that exhibits a transition from a uniform memory mechanism to a selective amnesic memory mechanism. Using a vector martingale approach, we study the asymptotic behaviour of the model across different parameter regimes. In the diffusive and critical regimes, we establish almost sure convergence, laws of the iterated logarithm, asymptotic normality of the walk, and the asymptotic rate of the mean square displacement. In the superdiffusive regime, we prove an almost sure convergence result and derive the corresponding mean square displacement rate. Also, we study the asymptotic behaviour of the center of mass associated with the walk. Later, we extend the model by incorporating random step sizes and obtain similar asymptotic results.

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 / 0 minor

Summary. The paper introduces a modified elephant random walk featuring a transition from uniform memory to selective amnesic memory via a specific functional form. It applies vector-martingale methods to derive almost sure convergence, laws of the iterated logarithm, asymptotic normality, and mean-square displacement rates in the diffusive and critical regimes; almost sure convergence and MSD rates in the superdiffusive regime; analogous results for the center of mass; and extensions to random step sizes.

Significance. If the vector-martingale representation holds uniformly, the work supplies a new parametric family of long-memory walks with an explicit memory-transition mechanism and a full suite of limit theorems across regimes, extending the existing elephant-random-walk literature with reproducible martingale-based derivations.

major comments (1)
  1. [Model definition / transition function] The abstract and model description state that all limit theorems follow from a vector-martingale representation whose validity hinges on the chosen transition function making the conditional increment have zero drift for every history. No explicit verification or cancellation identity is supplied in the provided text; this identity is load-bearing for every stated theorem and must be checked against the precise functional form in the model definition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a load-bearing gap in the presentation. We agree that an explicit verification of the zero-drift identity is necessary and will be supplied in the revision.

read point-by-point responses

  1. Referee: [Model definition / transition function] The abstract and model description state that all limit theorems follow from a vector-martingale representation whose validity hinges on the chosen transition function making the conditional increment have zero drift for every history. No explicit verification or cancellation identity is supplied in the provided text; this identity is load-bearing for every stated theorem and must be checked against the precise functional form in the model definition.

    Authors: We agree that the zero-drift condition must be verified explicitly for the specific transition function. The manuscript text supplied to the referee indeed omits this calculation. In the revised version we will add a short lemma (placed immediately after the model definition) that computes the conditional expectation of the increment and exhibits the exact cancellation that yields zero drift for every admissible history. This lemma will be referenced in the statements of all subsequent limit theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via model definition and martingale arguments

full rationale

The paper defines a specific transition function for the memory mechanism in the elephant random walk model and then applies vector-martingale techniques to derive almost-sure convergence, LIL, CLT, and MSD rates in the stated regimes. No quoted equations or self-citations reduce any claimed theorem to a fitted parameter, self-referential definition, or prior author result by construction. The martingale property follows directly from the chosen functional form as part of the model setup, after which the limit theorems are proved as consequences; this is a standard non-circular modeling approach with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a new stochastic process whose definition is the primary addition; it rests on the standard axioms of probability and the assumption that the memory transition preserves the martingale structure needed for the proofs.

axioms (1)
  • domain assumption The memory transition from uniform to selective amnesic preserves the vector-martingale property across parameter regimes.
    Invoked to justify application of martingale convergence theorems.

pith-pipeline@v0.9.1-grok · 5642 in / 1194 out tokens · 34045 ms · 2026-06-26T13:45:27.373205+00:00 · methodology

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Reference graph

Works this paper leans on

30 extracted references · 2 canonical work pages · 1 internal anchor

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