A study of renewal processes with infinite means (Q2471638)

Consider an ordinary renewal process with lifetimes \(T_i\) and an alternating renewal process with lifetimes \(T_i\) and preceding waiting times \(X_i\). It is assumed that \(EX_i= ET_i=\infty\). Then the processes admit limit theorems when the tails \(P\,(T_i\geq x)\) and \(P\,(X_i\geq x)\) vary regularly at \(\infty\) of index \(\alpha\in(0,1]\) and \(\beta\in(0,1]\). The authors are interested in the rate of convergence to these limits. Since this is a very difficult subject they take a numerical approach, useful for practical applications. One method is testing statistically whether the distribution of a random variable \(Y\) of the process at time \(t\) is approximately equal to its limit distribution or differs significantly from it. The distribution at time \(t\), for a number of \(t\)-values, was obtained by simulation. The second method is finite approximation. As the distribution of \(Y\) at time \(t\) take the limiting one with changed parameter(s) estimated by maximum likelihood, again by simulation. Here the test is by regression against \(t\) of the estimated parameters. The processes tested are: Ordinary renewal processes with \(P\,(T_i\geq x)= (1+ x)^{-\beta}\), \(0<\beta< 1\), with \(\beta= 1\), and with \(P\,(T_i\geq x)= \exp(-1/x)\). Alternating renewal processes with \(P\,(X_i\geq x)= (1+ x)^{-\alpha}\), \(P\,(T_i\geq x)= (1+ x)^{-\beta}\) with \(1/2<\alpha<\beta< 1\) and with \(1<2<\beta<\alpha< 1\). The approximations are found to be reasonable and best so in the interior of the parameter domain.