  Let $G$ be a finitely generated group of polynomial volume growth equipped
with a word-length $|\cdot|$. The goal of this paper is to develop techniques
to study the behavior of random walks driven by symmetric measures $\mu$ such
that, for any $\epsilon>0$, $\sum|\cdot|^\epsilon\mu=\infty$. In particular, we
provide a sharp lower bound for the return probability in the case when $\mu$
has a finite weak-logarithmic moment.
