Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 27J / c / Solution

The residual lifetime process, or excess process, isIf the inter-renewal distribution is non-arithmetic and $0<\mu =\mathbb{E}{\xi }_1<\infty$, then the renewal excess limit theorem saysEquivalently, $E_{\infty }$ has the law of $U\hat{\xi }$, where $U$ is uniform on $(0,1)$ and independent of the size-biased inter-renewal time $\hat{\xi }$.