Abstract
We consider a discrete-time Markov chain (Xt,Yt), t = 0, 1, 2,…, where the X-component forms a Markov chain itself. Assume that (Xt) is Harris-ergodic and consider an auxiliary Markov chain [Formula: see text] 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. We first provide natural conditions in terms of test functions ensuring that the [Formula: see text]-chain is positive recurrent and then prove that these conditions are also sufficient for positive recurrence of the original chain (Xt, Yt). The we prove a “multi-dimensional” extension of the result obtained. In the second part of the paper, we apply our results to two versions of a multi-access wireless model governed by two randomised protocols.
Highlights
We develop an approach to the stability analysis based on an averaging Lyapunov criterion
We further assume that the Markov chain {Xt} is Harris-ergodic, i.e. there exists a unique stationary distribution πX and, for any initial value X0 = x, the distribution of Xt converges to the stationary distribution in the total variation norm: sup |P(Xt ∈ A) − πX (A)| → 0 as t → ∞
The results of the previous section hold in this case too. They turn out not to be applicable in the examples we are going to consider in the second part of the paper, and we develop conditions only involving each individual coordinate of the Y -chain
Summary
We develop an approach to the stability analysis based on an averaging Lyapunov criterion. Under the conditions A and B there exists N0 such that the set D := V × {y : L2(y) ≤ N0} is positive recurrent for the Markov Chain {(Xt, Y t)}.
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