[Paper Review] Elephant random walks with delays

Allan Gut, Ulrich Stadtmüller|arXiv (Cornell University)|Jun 11, 2019

Stochastic processes and statistical mechanics8 references4 citations

TL;DR

This paper extends the elephant random walk (ERW) to include delays—allowing the walker to stay put with positive probability—by generalizing the memory-dependent step mechanism to a three-state process (±1, 0). It establishes a law of large numbers and asymptotic normality for the position, showing that the walk exhibits linear growth in expectation and variance, with limiting distributions depending on the memory structure and delay probability.

ABSTRACT

In the simple random walk the steps are independent, viz., the walker has no memory. In contrast, in the Elephant Random walk (ERW), which was introduced by Schütz and Trimper in 2004, the walker remembers the whole past, and the next step always depends on the whole path so far. One extension, as suggested in a recent paper by Bercu et al. (arXiv:1902.11220v1), is to allow for delays, that is, to put mass at zero. Our aim is to extend known result for the ordinary ERW to a Random Walk With Delays (ERWD).

Motivation & Objective

• To generalize the classical elephant random walk by incorporating the possibility of staying put (delays) at each step.
• To extend known results on the ordinary ERW—such as the law of large numbers and asymptotic normality—to the case with delays.
• To analyze the behavior of the walk under restricted memory regimes (e.g., remembering only the first step or the most recent step).
• To characterize the limiting distribution of the position process, including the mean and variance, under various memory and delay configurations.
• To identify phase transitions and critical parameters in the delayed ERW model, particularly in relation to the memory structure.

Proposed method

• Define the elephant random walk with delays (ERWD) by allowing the next step to be +X_K, -X_K, or 0 with probabilities p, q, and r (p+q+r=1), where K is uniformly chosen from 1 to n.
• Use the conditional expectation E(X_{n+1} | G_n) = (2p - 1) * S_n / n to derive martingale structures and moment equations.
• Apply the method of moments to compute the first and second absolute and central moments of the position T_n under fixed initial conditions.
• Derive asymptotic expressions for E(T_n) and Var(T_n) using difference equations and asymptotic expansions, showing linear growth in n.
• Establish weak convergence via uniform ergodicity of the Markov chain formed by the walk, enabling application of central limit theorems for Markov chains.
• Generalize results from fixed initial values (X_1 = 1, -1, or 0) to the random initial case by combining expectations and variances with mixture distributions.

Experimental results

Research questions

• RQ1 How does the inclusion of delays (i.e., a non-zero probability of staying put) affect the long-term behavior of the elephant random walk?
• RQ2 What are the asymptotic laws of large numbers and central limit theorems for the ERW with delays under full and restricted memory?
• RQ3 Does the phase transition at p = 3/4 in the standard ERW persist in the delayed ERW model, and how does the critical point depend on the delay probability r?
• RQ4 How do the mean and variance of the position grow over time when the elephant remembers only the first step or only the most recent step?
• RQ5 Can the limiting distribution of the normalized position be characterized as a mixture of normal and point masses when the initial step is random?

Key findings

• The expected position E(S_n)/n converges in probability to (p - q)/(2 + q - p) as n → ∞, establishing a law of large numbers for the delayed ERW.
• The variance of S_n grows linearly with n, with Var(S_n) ~ n * σ_S², where σ_S² = [(p - q)² / (2 + q - p)²] * (p + q - (p - q)²).
• The normalized position (S_n - n(p - q)X_1 / (2 + q - p)) / √n converges in distribution to a mixture of a normal distribution and a point mass at zero, specifically (p + q)N(0, σ_T²) + rδ₀.
• When the elephant remembers only the first step, the mean grows linearly with n, and the variance is also linear, with explicit expressions derived via moment equations.
• For the case of remembering only the most recent step, the process is Markovian and uniformly ergodic, enabling the application of central limit theorems for Markov chains.
• The critical point for phase transition in the delayed model is when p - q = 1/2, which reduces to p = 3/4 when r = 0, consistent with the original ERW.