On approximation of the analytic fixed finite time large.. probability distributions in an extreme renewal process with no-mean inter-renewals

We consider an extreme renewal process with no-mean heavy-tailed Pareto(II) inter-renewals and shape parameter.. where 0 < alpha <= 1. Two steps are required to derive integral expressions for the analytic probability density functions (pdfs) of the fixed finite time.. excess, age, and total life, and require extensive computations. Step 1 creates and solves a Volterra integral equation of the second kind for the limiting pdf of a basic underlying regenerative process defined in the text, which is used for all three fixed finite time t pdfs. Step 2 builds the aforementioned integral expressions based on the limiting pdf in the basic underlying regenerative process. The limiting pdfs of the fixed finite time t pdfs as t -> infinity do not exist. To reasonably observe the large t pdfs in the extreme renewal process, we approximate them using the limiting pdfs having simple well-known formulas, in a companion renewal process where inter-renewals are right-truncated Pareto(II) variates with finite mean; this does not involve any computations. The distance between the approximating limiting pdfs and the analytic fixed finite time large.. pdfs is given by an L-1 metric taking values in (0, 1), where "near 0" means "close" and "near 1" means "far".