Relative stability and weak convergence in non-decreasing stochastically monotone Markov chains (Q1180690)

Summary: Let \(\{\xi_ n\}\) be a non-decreasing stochastically monotone Markov chain whose transition probability \(Q(\cdot,\cdot)\) has \(Q(x,\{x\})=\beta(x)>0\) for some function \(\beta(\cdot)\) that is non- decreasing with \(\beta(x)\uparrow 1\) as \(x\to+\infty\), and each \(Q(x,\cdot)\) is non-atomic otherwise. A typical realization of \(\{\xi_ n\}\) is a Markov renewal process \(\{(X_ n,T_ n)\}\), where \(\xi_ j=X_ n\) for \(T_ n\) consecutive values of \(j\), \(T_ n\) geometric on \(\{1,2,\ldots\}\) with parameter \(\beta(X_ n)\). Conditions are given for \(X_ n\) to be relatively stable and for \(T_ n\) to be weakly convergent.