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A Ratio Limit Theorem for (Sub) Markov Chains on {1, 2, ···} with Bounded Jumps

Harry Kesten
Advances in Applied Probability
Vol. 27, No. 3 (Sep., 1995), pp. 652-691
DOI: 10.2307/1428129
Stable URL: http://www.jstor.org/stable/1428129
Page Count: 40
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
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Abstract

We consider positive matrices Q, indexed by {1,2,⋯ }. Assume that there exists a constant $1\leq L<\infty$ and sequences $u_{1}d_{r}+L>d_{r}>j$ 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 $\underset n\rightarrow \infty \to{\text{lim}}\frac{Q^{m+n}(i,j)}{Q^{n}(k,l)}=R^{-m}\frac{f(i)\mu (j)}{f(k)\mu (l)}$ 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 limn→ ∞ Qn+m(i,j){Σ ℓ Qn(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 μ for which R-1μ =μ Q). The results have an immediate interpretation for Markov chains on {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.

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