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

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 $(S_n{)}_{n\ge 1}$ 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 $(X_t{)}_{t\ge 0}$ 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.