Random walks with echoed steps I

TL;DR

This paper introduces a new type of random walk with memory, called RWES, which generalizes previous models and exhibits phase transitions and super-linear scaling, with detailed convergence and distributional results.

Contribution

It defines the RWES model, analyzes its convergence properties, phase transitions, and distributional limits, extending prior random walk models with memory.

Findings

Phase transition at p*E[ξ]=1.

Super-linear scaling in the super-critical regime.

Convergence to random series and distributional properties.

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 $\xi_n$. Namely, $\tilde{X}_n=\xi_n\tilde{X}_{\mathcal{U}[n]}$ 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 $\xi$ is non-negative. The rates of convergence are determined by the product $p\mathbb{E}\xi$ and exhibit a phase transition with critical value at $p\mathbb{E}\xi=1$. 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, P\'olya urns, and associated martingales.