Limit theorems for some long range random walks on torsion free nilpotent groups

TL;DR

This paper establishes limit theorems for long-range random walks on torsion-free nilpotent groups, showing convergence to stable-like processes on associated Lie groups, with results dependent on the measure used.

Contribution

It introduces a framework for deriving limit theorems for long-range random walks on torsion-free nilpotent groups, including the construction of a limit Lie group and process.

Findings

Convergence of rescaled random walks to stable-like processes on Lie groups.

Construction of a homogeneous nilpotent Lie group associated with the walk.

Establishment of local and functional limit theorems for these walks.

Abstract

We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $\Gamma$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $\mu$, we construct a homogeneous nilpotent Lie group $G_\bullet(\Gamma,\mu)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the measure $\mu$. In addition, the functional limit theorem is complemented by a local limit theorem.