Convergence of Random Walks in $\ell_p$-Spaces of Growing Dimension

TL;DR

This paper establishes a limit theorem for high-dimensional random walk paths viewed as metric spaces, showing convergence to a deterministic space as both steps and dimension grow, generalizing previous results for the Euclidean case.

Contribution

It extends convergence results of random walk paths in $ ext{l}_p$-spaces to arbitrary $p eq 2$, with growing dimension and steps, under specific moment and independence assumptions.

Findings

Random walk paths converge in probability to a deterministic limit space.

The convergence holds with respect to the Gromov-Hausdorff distance.

The result generalizes earlier work for the Euclidean case ($p=2$).

Abstract

We prove the limit theorem for paths of random walks with $n$ steps in $\mathbb{R}^d$ as $n$ and $d$ both go to infinity. For this, the paths are viewed as finite metric spaces equipped with the $\ell_p$-metric for $p\in[1,\infty)$. Under the assumptions that all components of each step are uncorrelated, centered, have finite $2p$-th moments, and are identically distributed, we show that such random metric space converges in probability to a deterministic limit space with respect to the Gromov-Hausdorff distance. This result generalises earlier work by Kabluchko and Marynych for $p=2$.