A ratio limit theorem for (sub) Markov chains on {1,2, .} with bounded jumps

We consider positive matrices Q, indexed by {1,2, .}. Assume that there exists a constant 1 L < . and sequences u1< u2< · ·· and d1d2< · ·· such that Q(i, j) = 0 whenever i < ur < ur + L < j or i > dr + L > dr > j for some r. If Q satisfies some additional uniform irreducibility and aperiodicity assumptions, then for s > 0, Q has at most one positive s-harmonic function and at most one s-invariant measure µ. We use this result to show that if Q is also substochastic, then it has the strong ratio limit property, that is https://static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aartic le%3AS0001867800027105/resource/name/S0001867800027105_eqn1.gif?pub-stat us=live for a suitable R and some R.1-harmonic function f and R.1-invariant measure µ. Under additional conditions µ can be taken as a probability measure on {1,2, .} and https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:bina ry:20180209072832852-0471:S0001867800027105:S0001867800027105_inline2.gi f?pub-status=live 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 µ for which R.1µ = µQ). 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.