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

Bochen Jin

Paper

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

Abstract

We prove the limit theorem for paths of random walks with

steps in

and

both go to infinity. For this, the paths are viewed as finite metric spaces equipped with the

-metric for

. Under the assumptions that all components of each step are uncorrelated, centered, have finite

-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