The paper considers positive matrices Q, indexed by . Assume that there exists a constant and sequences and such that whenever or for some r. If Q satisfies some additional uniform irreducibility and aperiodicity assumptions, then for . Q has at most one positive s-harmonic function and at most one s-invariant measure . The paper uses this result to show that if Q is also substochastic, then it has the strong ratio limit property, that is for a suitable and some -harmonic function f and -invariant measure . Under additional conditions can be taken as a probability measure on and exists. An example shows that this limit may fail to exist if Q does not satisfy the restrictions imposed above, even though Q may have a minimal normalized quasi-stationary distribution (i.e. a probability measure for which ). The results have an immediate interpretation for markov chains on with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditined on not yet being absorbed at 0 by time n.
