Random walks with echoed steps I

TL;DR

This work introduces the random walk with echoed steps (RWES), a memory-augmented process where, with probability $p$, increments follow an ordinary random walk step, and with probability $1-p$ they echo a past step scaled by a nonnegative random factor $\xi$. The authors establish a phase transition at $pm_1=1$, with supercritical behavior ($pm_1>1$) exhibiting super-linear growth and a nondegenerate random limit when $E[\xioldsymbol{log}\xi]<m_1$, while subcritical and critical regimes ($pm_1\le1$) yield LLNs around a mean with limiting random series representations. Central to the analysis are three representations: a continuous-time branching random walk (BRW), uniform random recursive trees (URRTs) with percolation, and associated Polya urns, all tied together by martingale methods. The paper provides precise limit theorems, moment conditions, and distributional identities for the limiting random variables, offering a robust framework for RWES and related memory-augmented processes with potential applications in economics and beyond.

Abstract

A random walk with echoed steps (RWES) is a process $\{\tilde{S}_n\}_{n\geq1}=\{\tilde{X}_1+\cdots+\tilde{X}_n\}_{n\geq1}$ that inserts memory and echo into an ordinary random walk (ORW) with i.i.d. steps, $X_1+\cdots+X_n$. The RWES is defined recursively as follows. Let $\tilde{S}_1=X_1$. With probability $1-p$, the $n$-th increment of the RWES follows that of the ORW, $\tilde{X}_n=X_n$. Otherwise, $\tilde{X}_n$ is set as a random echo of a uniform sample of the past steps $\tilde{X}_1,\dots,\tilde{X}_{n-1}$ determined by a random factor $ξ_n$. Namely, $\tilde{X}_n=ξ_n\tilde{X}_{\mathcal{U}[n]}$ with probability $p$, where $\mathcal{U}[n]\sim$Uniform$\{1,\dots,n-1\}$. The RWES is a broad generalization of the elephant random walk and of the positively/negatively/unbalanced step-reinforced random walks. We determine strong convergences of $\tilde{S}$ when the echo law $ξ$ is non-negative. The rates of convergence are determined by the product $p\mathbb{E}ξ$ and exhibit a phase transition with critical value at $p\mathbb{E}ξ=1$. Highlight that in its super-critical regime, the RWES has super-linear scaling exponents --observed for the first time in this type of random walks with memory--. We provide Laws of Large Numbers, conditions for the convergence of $\tilde{S}$ around its mean towards random series and provide some distributional properties of the limits. Our approach relies on the interpretation of the model in terms of continuous time branching random walks, random recursive trees, Pólya urns, and associated martingales.

Figures (1)

• Figure 2.1: Samples of the memory tree associated to the random walk with echoed steps $\tilde{S}$ for $p=1$ and $p\in(0,1)$.

Theorems & Definitions (21)

• proof : Proof.
• proof : Proof of Proposition \ref{['prop:||T(n)||ismtg']}
• proof : Proof.
• proof : Proof of Proposition \ref{['prop:convergencecasepeq1']}ii)
• proof : Proof.
• proof : Proof.
• proof : Proof of Proposition \ref{['prop:existenceQandconvW']}
• proof : Proof.
• proof : Proof.
• proof : Proof.