Failure rates of regenerative systems with heavy tails (Q1291185)

This paper is concerned with a regenerative process \(\xi_t\) which, for each state \(x\), can fail with intensity \(\alpha(x)\). If the inter-regeneration times have a finite exponential moment or \(\inf_x\alpha(x)> 0\), then it is known that usually \(\alpha(\xi_t)\) tends to a limiting positive intensity as \(t\to\infty\). This paper, on the other hand, provides a variety of examples where \(\alpha(\xi_t)\to 0\), in cases where the inter-regeneration time has no exponential moment and \(\inf_x\alpha(x)= 0\). Generally the speed of this convergence, which is conveniently studied using the failure-rate function, depends on both the tail of the inter-regeneration time and \(\alpha(x)\).