Random walks with echoed steps I

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.