Yang and Hall and Wellner initiated investigations of the asymptotic uniform behaviour of mean residual life (MRL) processes. They obtained results holding true over fixed and expanding compact subintervals of $[0, \infty)$.
In this exposition we study MRL processes over the whole positive half-line $[0, \infty)$. We describe classes of weight functions which enable us to establish the (a) strong uniform-over-$[0, \infty)$ consistency and (b)weak uniform-over-$[0, \infty)$ approximation of MRL processes. We give examples which show the necessity of employing weight functions in order to have (a) and (b), and prove the optimality of the weight function classes which we make use of. Extending our results concerning (b), we discuss constructions of asymptotic confidence bands for unknown MRL functions. The width of the obtained confidence bands is regulated by weight functions depending on the available information on the underlying distribution function.
"Mean residual life processes." Ann. Statist. 24 (4) 1717 - 1739, August 1996. https://doi.org/10.1214/aos/1032298292