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

2022/11/15 by Kabluchko, Zakhar, Marynych, Alexander · 2 citations
#60G50 #FOS: Mathematics #Primary: 60F05 #Probability (math.PR) #Secondary: 60D05

paper · doi:10.48550/arxiv.2211.08538

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 ℓ², converges in probability in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense to the Wiener spiral, as d,n→∞. Another group of results describes various possible limit distributions for the squared distance between the random walker at time n and the origin.

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• Convergence of Random Walks in ℓ_p-Spaces of Growing Dimension
• Ranges of Extremal Processes and Heavy-Tailed Random Walks in Spaces of Growing Dimension

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