Weibull renewal processes

We study the class of renewal processes with Weibull lifetime distribution from the point of view of the general theory of point processes. We investigate whether a Weibull renewal process can be expressed as a Cox process. It is shown that a Weibull renewal process is a Cox process if and only if 0<α≤1, where α denotes the shape parameter of the Weibull distribution. The Cox character of the process is analyzed. It is shown that the directing measure of the process is continuous and singular.