Association in time of a finite semi-Markov process (Q1914294)

Recall the definition: A stochastic process \(X(t)\), \(t \geq 0\) is associated in time, if for any \(n\) and \(t_1 < \dots < t_n\) and non-decreasing functions \(f\) and \(g\), we have \(\text{Cov} (f(T), g(T)) \geq 0\), where \(T = (X(t_1),\dots, X(t_n))\) and it is assumed that \(Ef(T)\), \(Eg(T)\), \(E[f(T)g(T)]\) exist. It is known that for complex repairable systems the assumption that the underlying stochastic process is associated in time allows to obtain the bounds for reliability of the system. So the problem of formulating corresponding sufficient condition arises. For the semi-Markov process with finite state space three equivalent conditions for association in time are given. They are expressed in terms of transition probabilities and crude hazard rates. Some examples are also considered.