The Prodigal Elephant Returns: Last-Return Laws in the Superdiffusive Regime
Abstract
Let (Sn) be the one-dimensional elephant random walk with memory parameter p∈(3/4,1) and first-step probability q=ℙ(S1=1)∈[0,1]. Set a=2p−1, and let fq,a denote the density of the almost-sure limit of Sn/na. We prove that the last return time T0=sup{m≥0:Sm=0} satisfies: ℙ(T0≥ n)∼ fq,a(0)√(8/(π(2a−1)))n−(a−1/2). Reflection gives fq,a(0)=f1,a(0), so the constant is independent of q. For future walks started from admissible states (n,xn) with xn/√n→ z, we establish functional convergence of the centered martingale-normalized paths to a time-changed Brownian motion. Uniform control of late hitting times then gives the eventual-return kernel 2Φ(−√(2a−1)|z|), uniformly on bounded scaled windows, where Φ is the standard normal distribution function. The same last-visit asymptotic holds uniformly for nonempty deterministic targets Fn⊆[−rn,rn]∩ℤ, rn=o(√n), held fixed during each trajectory. Conditionally on T0≥ n, we identify the joint limit of Sn/√n, the next return divided by n, and the last return divided by the next return. The last ratio converges to A−1/(2a−1), independently of the other two limiting coordinates, where A has the arcsine distribution on (0,1).
Related
- A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity
- From Discrete Trailing Returns to a Continuous Graphical Profile: Return-to-Present Curves
- Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift
- Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension
- Superdiffusive Invariance Principle for the two-dimensional Asymmetric Simple Exclusion Process
BibTeX
@misc{trinas2026prodigal,
title = {{The Prodigal Elephant Returns: Last-Return Laws in the Superdiffusive Regime}},
author = {Hélio Trinas},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01279},
eprint = {2611.01279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01279},
}
RIS
TY - UNPB
TI - The Prodigal Elephant Returns: Last-Return Laws in the Superdiffusive Regime
AU - Hélio Trinas
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01279
UR - https://arxiv.org/abs/2611.01279
AB - Let (S_n) be the one-dimensional elephant random walk with memory parameter p∈(3/4,1) and first-step probability q=ℙ(S_1=1)∈[0,1]. Set a=2p−1, and let f_q,a denote the density of the almost-sure limit of S_n/n^a. We prove that the last return time T_0=sup{m≥0:S_m=0} satisfies: ℙ(T_0≥ n)∼ f_q,a(0)√(8/(π(2a−1)))n^−(a−1/2). Reflection gives f_q,a(0)=f_1,a(0), so the constant is independent of q. For future walks started from admissible states (n,x_n) with x_n/√n→ z, we establish functional convergence of the centered martingale-normalized paths to a time-changed Brownian motion. Uniform control of late hitting times then gives the eventual-return kernel 2Φ(−√(2a−1)|z|), uniformly on bounded scaled windows, where Φ is the standard normal distribution function. The same last-visit asymptotic holds uniformly for nonempty deterministic targets F_n⊆[−r_n,r_n]∩ℤ, r_n=o(√n), held fixed during each trajectory. Conditionally on T_0≥ n, we identify the joint limit of S_n/√n, the next return divided by n, and the last return divided by the next return. The last ratio converges to A^−1/(2a−1), independently of the other two limiting coordinates, where A has the arcsine distribution on (0,1).
ER -