Random Walks in the High-Dimensional Limit I: The Wiener Spiral

TL;DR

This paper establishes limit theorems for high-dimensional random walks, showing convergence to the Wiener spiral and analyzing the distribution of squared distances as both steps and dimension grow large.

Contribution

It introduces new limit theorems describing the asymptotic behavior of high-dimensional random walks and their geometric properties.

Findings

Path converges to the Wiener spiral in high dimensions.

Limit distributions for squared distances are characterized.

Convergence occurs in Hausdorff and Gromov-Hausdorff senses.

Abstract

We prove limit theorems for random walks with $n$ steps in the $d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of our results states that the path of such a random walk, viewed as a compact subset of the infinite-dimensional Hilbert space $\ell^2$, converges in probability in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense to the Wiener spiral, as $d,n\to\infty$. Another group of results describes various possible limit distributions for the squared distance between the random walker at time $n$ and the origin.