Convergence of Random Walks in $ll_p$-Spaces of Growing Dimension

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 $p=2$.

💡 Research Summary

The paper studies the asymptotic geometry of random walk paths when both the number of steps n and the ambient dimension d tend to infinity. Each step of the walk is a d‑dimensional vector whose coordinates are independent, identically distributed, centered random variables ξ₁,…,ξ_d with variance σ² and finite 2p‑th moment, scaled by d^{-1/p} so that the ℓₚ‑norm of a single step has unit order. The path Zₙ^{(d)}={S₀^{(d)},…,Sₙ^{(d)}} is regarded as a finite metric space equipped with the ℓₚ‑metric, and the authors investigate its convergence in the Gromov–Hausdorff sense.

The main result (Theorem 1.1) asserts that the rescaled metric space (n^{-1/2}Zₙ^{(d)},‖·‖ₚ) converges in probability, as n→∞ and d=d(n)→∞, to a deterministic limit space: the unit interval