Failure rates of regenerative systems with heavy tails

Let ξt be a regenerative process and assume that, at each state x, the process can fail with intensity α(x). If the inter-regeneration times have a finite exponential moment orinfxα(x)>0, then α(ξt) tends to some limiting positive intensity as t→∞ (under mild additional restrictions). This fact is widely used in engineering because the limiting intensity can be employed in various calculations, say, in reliability theory. The paper contains a variety of examples showing that α(ξt) provided that inter-regeneration time has no exponential moment andinfx α(x)=0. The speed of convergence depends, in general, on both the tail of inter-regeneration time and α(x).