We consider a discrete-time Markov chain (Xn, Yn), where the X-component forms a Markov chain itself. Assuming that (Xn) is ergodic, we formulate the following "naive" conjecture. Consider an auxiliary Markov chain whose transition probabilities are the averages of transition probabilities of the Y-component of the (X, Y)-chain, where the averaging is weighted by the stationary distribution of the X-component. The conjecture is: if the this chain is positive recurrent, then so is the (X, Y)-chain. We first show that, under appropriate technical assumptions, such a general result indeed holds, and then apply it to two versions of a multi-access wireless model governed by two randomised protocols.
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