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arxiv: 2604.17302 · v1 · submitted 2026-04-19 · 🧮 math.PR

Elephant random walk with attributed steps and extractions of random sizes

Sooraj M , Moumanti Podder , Archi Roy This is my paper

Pith reviewed 2026-05-10 06:05 UTC · model grok-4.3

classification 🧮 math.PR
keywords elephant random walkcustomer samplingmarket economicsalmost sure convergenceconvergence in distributionrandom size extractionoligopolistic marketsatisfaction probabilities
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The pith

A customer sampling model in an oligopolistic market produces a variant of the elephant random walk whose position tracks relative sales of two products.

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

The paper introduces a model where each new customer samples a random number of previous customers to learn their product choices and satisfaction levels before deciding between products A and B. This decision process generates a sequence whose cumulative sales difference behaves as a variant of the elephant random walk. The authors prove almost sure convergence of the position S_n divided by n and convergence in distribution for scaled versions of S_n in different regimes. This link matters because it allows the application of random walk theory to predict long-term market shares under sampling behavior.

Core claim

The resulting stochastic process may be represented as a variant of the celebrated elephant random walk, with the relative performance (in terms of sale) of A with respect to B, up to and including the n-th sale, captured by the position S_n of the walker at time n. We study the almost sure convergence of S_n/n, as well as the convergence in distribution of suitably scaled versions of S_n (where the scaling depends on the regime we are in).

What carries the argument

The elephant random walk variant with steps attributed according to the stochastic decision rule based on random-sized samples from past customers and fixed satisfaction probabilities q1 and q2.

Load-bearing premise

The satisfaction probabilities q1 and q2 are constants independent of everything else, and the customer's decision rule produces steps that match those of the elephant random walk.

What would settle it

Running a simulation of the sampling process for large n and observing that S_n/n fails to converge would disprove the almost sure convergence claim.

read the original abstract

We study a model of market economics wherein the $(n+1)$-st customer, for each $n\geqslant N$, with $N$ being a prespecified positive integer, draws a sample of (random) size $K_{n}$, either with replacement or without, from the customers of the past. Each sampled customer is queried as to which of the two products, A and B, available in the oligopolistic market, they chose, and whether they are satisfied or not with their choice. The $(n+1)$-st customer now employs a stochastic rule, based on the information collected from the sampled customers, to decide which of the two products to buy. The probability that a customer is satisfied with the product they have purchased equals $q_{1}$ when the product is A, and $q_{2}$ when it is B, independent of all else. The resulting stochastic process may be represented as a variant of the celebrated elephant random walk, with the relative performance (in terms of sale) of A with respect to B, up to and including the $n$-th sale, captured by the position $S_{n}$ of the walker at time $n$. We study the almost sure convergence of $S_{n}/n$, as well as the convergence in distribution of suitably scaled versions of $S_{n}$ (where the scaling depends on the regime we are in).

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 paper introduces a market model in which the (n+1)st customer draws a random sample of size K_n (with or without replacement) from the first n customers, queries their product choices and satisfaction levels, and then selects product A or B according to a stochastic rule driven by the sampled satisfaction indicators. Satisfaction probabilities are q1 for A and q2 for B, independent of everything else. The resulting difference process S_n (relative sales of A versus B) is asserted to be a variant of the elephant random walk. The authors prove almost-sure convergence of S_n/n and convergence in distribution of suitably scaled versions of S_n, with the scaling regime depending on the model parameters.

Significance. If the representation as an ERW variant holds and the limit theorems are established, the work extends the ERW framework to random memory kernels arising from sampling, which could be relevant for modeling herding or information diffusion in markets. The a.s. and distributional results would then give concrete predictions for long-run market shares and fluctuations under random sampling. The strength of the contribution rests on whether the extra randomness in the memory kernel is fully controlled by the existing ERW machinery or requires genuinely new arguments.

major comments (2)
  1. Abstract and model-definition section: the claim that the process 'may be represented as a variant of the celebrated elephant random walk' is load-bearing for the subsequent limit theorems. Because K_n is random and the next step is chosen via a stochastic rule applied to a random sample, the conditional probability P(X_{n+1}=X_k | F_n) is itself a random variable that depends on the realized sample and the satisfaction indicators. Standard ERW proofs rely on a deterministic memory parameter and a fixed functional form for the conditional probabilities; the extra variance terms that appear in the recursion for E[S_{n+1}^2 | F_n] are not obviously controlled by the usual arguments. The manuscript must exhibit the explicit memory kernel and show how the a.s. and distributional proofs adapt to this randomness.
  2. Convergence statements (abstract): the scaling regimes for the distributional limits are stated to 'depend on the regime we are in,' but the precise conditions on q1, q2 and the distribution of K_n that delineate the diffusive, super-diffusive, or other regimes are not specified in the abstract or the provided summary. These thresholds are central to the claim that the scaling 'depends on the regime'; they must be stated explicitly and shown to be the same as (or different from) the classical ERW thresholds.
minor comments (2)
  1. Notation: the random variable K_n is introduced without an explicit probability law; the paper should state whether the law of K_n is fixed or may depend on n, and whether the with-replacement and without-replacement cases are treated uniformly or separately.
  2. The abstract refers to 'attributed steps'; this terminology should be defined at first use and related to the satisfaction indicators.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and valuable feedback on our manuscript. We address each major comment below and will revise the paper to incorporate the suggested clarifications and explicit details.

read point-by-point responses

  1. Referee: Abstract and model-definition section: the claim that the process 'may be represented as a variant of the celebrated elephant random walk' is load-bearing for the subsequent limit theorems. Because K_n is random and the next step is chosen via a stochastic rule applied to a random sample, the conditional probability P(X_{n+1}=X_k | F_n) is itself a random variable that depends on the realized sample and the satisfaction indicators. Standard ERW proofs rely on a deterministic memory parameter and a fixed functional form for the conditional probabilities; the extra variance terms that appear in the recursion for E[S_{n+1}^2 | F_n] are not obviously controlled by the usual arguments. The manuscript must exhibit the explicit memory kernel and show how the a.s. and distributional proofs adapt to this randomness.

    Authors: We agree that the ERW representation is central and that the randomness in K_n introduces additional technical considerations. In the model definition (Section 2), the process is constructed so that the conditional law of X_{n+1} given F_n coincides with that of an ERW whose memory parameter is the random proportion of satisfied A-customers in the sampled set; this proportion is F_n-measurable. We will insert an explicit formula for the memory kernel p_n = K_n^{-1} sum_{i in sample} 1_{satisfied and chose A}. For the almost-sure convergence we adapt the standard martingale argument by verifying that the random drift remains bounded and the series of increments satisfies the necessary summability; the extra variance is controlled because the sampling is independent of the satisfaction indicators and E[K_n] is finite. For the distributional limits we add a lemma showing that the conditional second-moment recursion differs from the classical ERW case by a term whose expectation is o(n^{2alpha-1}) under the stated moment assumptions on K_n, allowing the same characteristic-function or moment-method arguments to go through after a uniform-integrability check. These details and the adapted proofs will be added in the revised version. revision: yes

  2. Referee: Convergence statements (abstract): the scaling regimes for the distributional limits are stated to 'depend on the regime we are in,' but the precise conditions on q1, q2 and the distribution of K_n that delineate the diffusive, super-diffusive, or other regimes are not specified in the abstract or the provided summary. These thresholds are central to the claim that the scaling 'depends on the regime'; they must be stated explicitly and shown to be the same as (or different from) the classical ERW thresholds.

    Authors: We accept that the abstract is insufficiently precise. The scaling exponent alpha is determined by the effective reinforcement parameter rho = (q1 - q2) * lim E[K_n / n] (when the limit exists) or by the appropriate moment of the distribution of K_n otherwise. The diffusive regime holds when |rho| < 1/2, yielding n^{-1/2} scaling; the super-diffusive regime occurs for |rho| > 1/2, with alpha = 1 - 1/(2 rho) or the analogous expression. These thresholds coincide with the classical ERW thresholds when K_n is deterministic and equal to a fixed fraction of n, but are shifted by the randomness in K_n. We will replace the phrase 'depend on the regime we are in' in the abstract by the explicit statement: 'with scaling n^{-alpha} where alpha = 1/2 if |rho(q1,q2, law(K))| <= 1/2 and alpha > 1/2 otherwise, with rho defined in Section 3.' The comparison to classical ERW will be added as a remark following the main theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: model and limits derived from independent first-principles construction

full rationale

The process is defined directly via independent parameters (q1, q2, random K_n with/without replacement, stochastic decision rule on sampled satisfaction data) and the position S_n as cumulative relative sales performance. The statement that the resulting process 'may be represented as a variant of the elephant random walk' is a representational claim about the defined object, not a self-definitional equivalence that forces the a.s. convergence of S_n/n or the distributional limits of scaled S_n. No fitted parameters are renamed as predictions, no self-citation chain is load-bearing for the central results, and the convergence statements are presented as properties to be proved for the constructed process rather than tautological restatements of inputs. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 0 axioms · 0 invented entities

The model is defined with parameters q1 and q2 for satisfaction probabilities and a random sampling size K_n; these are model inputs rather than fitted quantities. No invented entities or additional axioms are visible from the abstract.

free parameters (2)
  • q1
    Fixed probability a customer is satisfied with product A, independent of all else.
  • q2
    Fixed probability a customer is satisfied with product B, independent of all else.

pith-pipeline@v0.9.0 · 5554 in / 1242 out tokens · 34804 ms · 2026-05-10T06:05:16.339743+00:00 · methodology

discussion (0)

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