Ergodicity and inequalities in a class of point processes
The behaviour of a renewal process in the future depends on the past only through the time that has elapsed since the last renewal. Here we have a class of point processes where the future depends on the past only through the last m points and those points no more than A time units into the past. (Those past points relevant to the future constitute what is here called the memory process). Under fairly strong conditions on the (history dependent) hazard rate for the next point an analogue of Blackwell's renewal theorem is established for these processes. The proof is done by establishing the Harris recurrence of the memory process. In an earlier paper [Stochastic Processes Appl. 22, 1-15 (1986; Zbl 0611.60086)], the author explored the consequences of assuming a decreasing failure rate in renewal theory. Here a coupling is established linking together two point processes with different histories and this, together with an extended notion of decreasing failure rates, yields some similar results in this more general setting.
scientific article; zbMATH DE number 4072049