On a theorem of Avez

For each symmetric, aperiodic probability measure $\mu$ on a finitely generated group $G$, we define a subset $A_{\mu}$ consisting of group elements $g$ for which the limit of the ratio ${\mu^{\ast n}(g)}/{\mu^{\ast n}(e)}$ tends to $1$. We prove that $A_\mu$ is a subgroup, is amenable, contains every finite normal subgroup, and $G=A_\mu$ if and only if $G$ is amenable. For non-amenable groups we show that $A_\mu$ is not always a normal subgroup, and can depend on the measure. We formulate some conjectures relating $A_\mu$ to the amenable radical.