On some random walks driven by spread-out measures

Let $G$ be a finitely generated group equipped with a symmetric generating $$-tuple $S$. Let $|\cdot|$ and $V$ be the associated word length and volume growth function. Let $\nu$ be a probability measure such that $$. We prove that if $G$ has polynomial volume growth then $\nu^{(n)}(e) \simeq V(\sqrt{n\log n})^{-1}$. We also obtain assorted estimates for other spread-out probability measures.