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