Exponentials and $R$-recurrent random walks on groups

On a locally compact group $E$ with a countable base we consider a right random walk $X$ which for some $r>0$ has a unique (up to a positive multiplier) $r$-invariant measure. If this measure obeys some weak restrictions, then the random walk $X$ corresponds to the single continuous exponential on $E$. From this we obtain that we can implement some $R$-recurrent (by Tweedie) random walk on the group $E$ only in the case when this group is recurrent and, moreover, when there exists a Harris recurrent random walk on it.