  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.
