Abstract
In this paper we find conditions under which ergodic Markov chains are also geometrically ergodic: that is, converge to their limits geometrically quickly. We show that if the increment distributions of the chain have uniform exponential tails in an appropriate sense, then the stronger geometric ergodicity hold, whilst if the stationary measure π of the chain has suitably exponential tails then again geometric ergodicity holds but under further auxiliary conditions. We give examples to show that, in particular, π may have geometric tails but the chain need not be geometrically ergodic. We conclude with a number of examples from queueing and network theory covered by the results, indicating the use of the results when there is a known Foster-Lyapunov function and also when the hitting times of finite sets are merely
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