Techniques for establishing ergodic and recurrence properties of continuous‐valued markov chains

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Abstract
We present techniques for classifying Markov chains with a continuous state space as either ergodic or recurrent. These methods are analogous to those of Foster for countable space chains. The theory is presented in the first half of the paper, while the second half consists of examples illustrating these techniques. The technique for proving ergodicity involves, in practice, three steps: showing that the chain is irreducible in a suitable sense; verifying that the mean hitting times on certain (usually bounded) sets are bounded, by using a “mean drift” criterion analogous to that of Foster; and finally, checking that the chain is such that bounded mean hitting times for these sets does actually imply ergodicity.The examples comprise a number of known and new results: using our techniques we investigate random walks, queues with waiting‐time‐dependent service times, dams with general and random‐release rules, the s‐S inventory model, and feedback models.

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