The asymptotic behaviour of the residual life time at time $t$ is investigated (for $t \rightarrow \infty$). We derive weak limit laws and their domains of attraction and treat rates of convergence and moment convergence. The presentation exploits the close similarity with extreme value theory.
"Residual Life Time at Great Age." Ann. Probab. 2 (5) 792 - 804, October, 1974. https://doi.org/10.1214/aop/1176996548