  Using the Perron-Frobenius eigenfunction and eigenvalue, each finite
irreducible nonnegative matrix $A$ can be transformed into a probability kernel
$P$. This was generalized by David Vere-Jones who gave necessary and sufficient
conditions for a countably infinite irreducible nonnegative matrix $A$ to be
transformable into a recurrent probability kernel $P$, and showed uniqueness of
$P$. Such $A$ are called R-recurrent. Let us say that $A$ is strongly
R-positive if the return times of the Markov chain with kernel $P$ have
exponential moments of some positive order. Then it is known that strong
R-positivity is equivalent to the property that lowering the value of finitely
many entries of $A$ lowers the spectral radius. This paper gives a short and
largely self-contained proof of this fact.
