Logarithmic scaling of planar random walk's local times

TL;DR

This paper demonstrates that when the local times of a planar simple random walk are scaled logarithmically, they converge to a non-degenerate jump process in the Skorokhod M1 topology, revealing new scaling behavior.

Contribution

It establishes the convergence of the scaled local time process to a pure jump process in the logarithmic time scale, highlighting a novel limit behavior in planar random walks.

Findings

Convergence in the Skorokhod M1 topology

Failure of convergence in the J1 topology

Identification of a non-degenerate jump process as the limit

Abstract

We prove that the local time process of a planar simple random walk, when time is scaled logarithmically, converges to a non-degenerate pure jump process. The convergence takes place in the Skorokhod space with respect to the $M1$ topology and fails to hold in the $J1$ topology.