Relative stability and weak convergence in non-decreasing stochastically monotone Markov chains. (1991) - Todorovic, P.
Abstract : Let ξ n 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 ξ n is a Markov renewal process ( X n, T n ) , where ξ j = X n, for T n consecutive values of j, T n geometric on 1, 2, … with parameter β ( X n ) . Conditions are given for X n, to be relatively stable and for T n to be weakly convergent.