Convergence of Random Walks in ${\mathrm{\ell }}_p$-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 ${\mathrm{\ell }}_p$-metric for $p\in [1,\mathrm{\infty })$. 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 $p=2$.