Residual lifetime process

The residual lifetime is the time remaining until the next renewal.

For non-arithmetic inter-renewal times with finite positive mean $\mu$, the excess $E(t)=S_{N(t)+1}-t$ converges in distribution to $E_{\infty }$ with

If an independent superposition of a Poisson process and a non-arithmetic finite-mean renewal process is itself a renewal process, then its first interarrival is exponential. Equating the limiting excess law computed as a renewal excess with the minimum of the Poisson and second-process excesses yields an integral equation whose survival-function solutions are exponential.

With bounded integer lifetimes, residual lifetime decreases deterministically and resets randomly at zero.

The invariant residual-life mass at i is proportional to the probability that a lifetime exceeds i.