Abstract
Let {ξn} be a non‐decreasing stochastically monotone Markov chain whose transition probability Q(..) has Q(x, {x}) = β(x) > 0 for some function β(.) that is non‐decreasing with β(x)↑1 as x → +∞, and each Q(x.) is non‐atomic otherwise. A typical realization of {ξn} is a Markov renewal process {(Xn, Tn)}, where ξj = Xn, for Tn consecutive values of j, Tn geometric on {1, 2, …} with parameter β(Xn). Conditions are given for Xn, to be relatively stable and for Tn to be weakly convergent.
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