We consider positive matrices Q, indexed by {1, 2,...}. Assume that there exists a constant 1 less than or equal to L < infinity and sequences u(1) < u(2) < and d(1) < d(2) ... such that Q(i,j) = 0 whenever i < u(r) < u(r) + L < j or i > d(r) + L > d(r) > j for some r. If Q satisfies some additional uniform irreducibility and aperiodicity assumptions, then for a > 0, Q has at most one positive s-harmonic function and at most one s-invariant measure mu. We use this result to show that if Q is also substochastic, then it has the strong ratio limit property, that is GRAPHICS for a suitable R and some R(-1)-harmonic function f and R(-1)-invariant measure mu. Under additional conditions mu can be taken as a probability measure on {1,2,...} and lim(n-->)Q(n+m)(i,j){Sigma(l)Q(n)(k,l)}(-1) 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 mu for which R(-1)mu = mu Q). The results have an immediate interpretation for Markov chains on {0,1,2,...} with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditioned on not yet being absorbed at 0 by time n.
